^^a^ 


.^M^^^^ 


University  of  California  •  Berkeley 

The  Theodore  P.  Hill  Collection 

of 
Early  American  Mathematics  Books 


OOlTCISEl 

Mercantile  Arithmetic, 

FOR 

COMMERCIAL  COLLEGES, 

AND    A 

Hand-Book  for  the  Counting-Room, 


CONTAINING  ALL  THE  MORE  USEFUL  AND   PRACTICAL  CALCU- 
LATIONS   OF    EVERY-DAY  APPLICATION,  EXPLAINED 
ON  SCIENTIFIC  PRINCIPLES. 


HENRY  A.  FABER, 

AUTHOR    OF    "THE   STATISTICAL    ACCOUNT-BOOK' 
AND    <'FABER'S   manual." 


THIRD  EDITION.— REVISED. 


CINCINNATI,   O.  : 

Queen  City  Commercial  College, 

N.  W.  Cor.  Fifth  and  Walnut  Streets. 
ATLANTA,  GA. : 

Moore's  Southern  Business    University. 
1880. 


Entered,  according  to  Act  of  Congress,  in  the  year  1876, 
BY  HENRY  A.  FABER, 

In  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


A//  rights  reserved. 


PREFACE. 


This  treatise  has  been  prepared  with  special  refer- 
ence to  the  wants  of  students  of  commercial  colleges. 
All  questions  which  tend  to  perplex  the  learner,  with 
little  or  no  practical  utility,  have  been  carefully  ex- 
chided. 

Rules  have  been  almost  altogether  omitted.  The 
student  must  see  the  operation,  and,  having  seen  it, 
his  judgment  will  enable  him  to  deduce  some  method 
of  solution  for  himself 

The  subjects  are  treated  of  in  the  order  of  their 
simplicity  and  utility.  They  are  so  independent  of 
each  other,  however,  that  the  teacher  may  introduce 
them  in  whatever  order  his  judgment  may  dictate. 

Numerous  exercises  will  be  found  on  short  methods 

of  calculation.     In  fact,   every  topic  which  admits  of 

more  than  one  form  of  solution,  has  been  treated  by 

the  shortest  practical  method. 

The  Author. 
Cincinnati,  1880. 


CONTENTS: 

TABLE  OF  MONEYS,  WEIGHTS,  MEASURES,  etc..  Page  6. 
ARITHMETIC. 


Chapter. 

1.  Introduction  .   .  . 

Page. 

.  15 

2 

Notation  and      .   . 

, 

Numeration  17 

3. 

Addition 

.  19 

4. 

Subtraction    .    .    . 

.  25 

5. 

Multiplication  .    . 

.28 

6. 

Division 

.36 

7- 

Easy  Fractions  .   . 

.43 

8. 

Decimals      

.53 

9. 

Short  Methods  .    . 

. 

OF    Multiplication 

and    Division  .  .   . 

.55 

10. 

Percentage  .... 

.61 

11. 

Bills-Invoices  .  .    . 

.67 

12. 

Long  Division    ,   . 

.  78 

13. 

Time 

.82 

14. 

Simple  Interest    , 

.83 

15. 

Compound     ,  .    .   . 

.  90 

16. 

Annual         .... 

.  91 

17. 

Partial  Payments  . 

.92 

Page. 
.    .94 


Chapter. 

18.  Bank  Discount 

19.  True  Discount  .   .  .  .99 

20.  DiscT.  Interest  Bear- 

ing Notes  .       .  100 
21  Complex  Percentage  103 

22.  Time  Tables  ....  114 

23.  Average      118 

24.  Ratio 138 

25.  Proportion 139 

26.  Partnership  ....  143 

27.  Joint  Stock  Co  .  .    .  146 

28.  Compound  Numbers  148 

29.  Foreign  Exch.      .   .  151 

30.  Importing 157 

31.  Farming 165 

32.  Lumber  Measure    .   169 

33.  Fractions 171 

34.  Duodecimals     .    .    .  190 

35.  CoMP.  Proportion    .  193 

36.  Gauging      195 


INDEX,  197  to  200  inclusive. 


MONEYS,  WEIGHTS  AND  MEASURES- 


MONEYS. 

Federal  Money. — The  unit  of  our  money  is  the  dollar. 
Accounts  are  kept  in  dollars  and  cents.  The  coins  are,  the 
double-eagle,  eagle,  half-eagle,  quarter-eagle,  and  dollar;  the 
trade  dollar,  half-dollar,  quarter  dollar,  dime,  and  half-dime; 
the  three,  five,  two,  and  one  cent  pieces.  Federal  money 
being  decimal  currency,  ten  of  a  lower  denomination  make 
one  of  a  higher:  10  mills  =  1  cent,  10  cents  =r  1  dime,  10 
dimes  =  1  dollar,  10  dollars -=  1  eagle.  Signs:  m,  mills; 
^,  cents;  $,  dollars;  E,  eagle. 

British  Money. — The  unit  of  British  money  is  the  pound 
sterling.  Accounts  are  kept  in  pounds,  shillings  and  pence 
(farthings  are  written  as  fractions  of  a  penny):  4  farthings 
=  1  penny,  12  pence  =  1  shilling,  20  shillings  =  1  pound. 
Signs:  d,  pence;  s,  shilling;  £,  pound.  The  British  coins 
are,  the  penny,  the  shilling,  the  crown,  the  sovereign,  and  the 
guinea.  The  value  of  the  crown  is  5  shillings;  the  sovereign, 
20  shillings;  the  guinea,  21  shillings. 

German  Money. — The  unit  of  the  money  of  the  German 
Empire  is  the  rixmark.  10  pennies  (pfennige)  =  1  silver- 
gropchen  (silbergroschen) ;  10  silver-groschen  =  1  rixmark 
(reichsmark).  Signs:  d,  pennies;  sg,  silver-groschen;  Rm, 
rixmark. 

French  Money. — The  unit  of  French  money  is  the  ifranc. 
10  centimes  =  1  decime;  10  decimes  =^  1  franc.  Signs:  c, 
centimes;  d,  decimes;  fc,  francs. 


MONEYS,  WEIGHTS   AND   MEASURES. 


WEIGHTS. 

Mint  or  Troy  Weight,  used  at  the  mint,  and  by  jewel- 
ers: 24  grains  =  1  pennyweight,  20  pennyweights  =  1  ounce, 
12  ounces  =  1  lb.    Signs,  gr.  grain,  pwt.  pennyweight,  oz.  ounce. 

Apothecaries'  Weight.— Used  in  compounding  medi- 
cines :  20  grains  =  1  scruple,  3  scruples  =  1  drachm,  8  drachms 
=  1  ounce,  12  ounces  =  1  lb.  Sighs,  gr.  grain,  9  scruple,  3 
drachm,  g  ounce,  R)  pound.  (1  lb.=  5760  gr.) 

Commercial  Weight,  used  by  Grocers,  Druggists,  Hard- 
ware dealers,  etc :  16  ounces  ==  1  pound,  2000  pounds  =  1  tun. 
Signs,  oz.  ounces,  lbs.  pounds,  cwt.  hundreds,  T  tuns.  (1  lb.= 
7000  gr.) 

Hay  is  weighed  by  Commercial  Weight 

Avoirdupois — Old  Commercial  Weight  of  the  U.  States 
16  drachms  =  1  ounce,  16  ounces  =  1  lb.,  28  lbs.  =  1  quarter 
4  quarters  =  1  hundred,  20  cwt.  ^=  1  tun. 

Pig  Iron  (chill  mold),  Iron  Ore,  Bituminous  Coal,  and  Hemp  are 
weighed  by  avoirdupois  weight. 

The  Avoirdupois  weiight  is  the   Commercial  weight  of  Grea 

BHtain. 

Metric  Weight.— The  unit  of  weights  of  the  metric  sys 
tem  is  the  Gram.  The  Greek  prefixes  (deka,  10;  Jiedo,  100; 
kilo,  1000)  form  the  denominations  above  tlie  unit.  The  Latin 
prefixes  [deci,  10;  centi,  100;  milliy  1000)  form  the  denomina- 
tions below  the  unit.  1  kilogram  =  10  hectogram  =  lOOfdeca- 
grams  =  1000  grams.  1  milligram  =  -^-^  centigram  =  -^ 
decigram  =  y^Vff  g^^^*  The  weight  of  a  gram  is  equal  to 
15.432  grains  of  Troy  weight.  Signs:  DG.,  deckagram;  HG., 
hectogram;  KG.,  kilogram;  G.,  gram;  dg.,  decigram;  eg., 
centigram;  mg.,  milligram.  . 

Note. — The  oz.  of  the  Mint  and  Apoth.  weights  are  the 
same,  viz:  480  grains.  The  oz.  of  the  Com'l  and  Avoid. 
•weights  are  the  same,  viz:  437 J  grains. 


MONEYS,   WEIGHTS    AND    MEASURES. 


Weights  op  Produce  per  Bushel,  according  to  usage 
in  Cincinnati,  and  as  fixed  by  statute  in  Ohio: 


Psage. 
lbs. 

Apples,  dried 26 

Barley 48 

Barley  malt,  weight  of 

bags  included 34 

Beans 60 

Bran 20 

Bran  shorts 25 

Broom-corn 30 

Buckwheat 60 

Coal,  bituminous 80 

cannel 70 

Charcoal 30 

Coke 32 

Castor  beans 46 

Corn,  shelled 66 

in  ear.. ..68  and  70 

Hair,  plastering 8 

wet 16 

Hominy 60 

Lime,  slacked 51 

Malt 

Meal,  corn 60 

Middlings 40 

Oats.... 32 

Onions 66 

Onion  sets 23 


Stat.  Usage.  Stat, 

lbs.  lbs.   lbs. 

25      Peaches,  dried 38     33 

48      Peas 60     60 

green 24 

Plaster  and  hair 118 

Peanuts,  roasted 22 

Potatoes,  Irish 60     60 

sweet 60 

Rye..... 66     66 

Rye  malt,  wt.  of  bags 

included  40 

Salt 66 

Seed,  clover 60    60 

timothy 46     45 

flax 66     66 

hemp 44     44 

orchard  grass....  14 

Hungarian  grass  50     60 

blue  grass 14 

millet 60    50 

canary  60 

sorghum 46 

Ship  stuff 40 

i  Shorts 30 

32  ;  Turnips 60 

j  Wheat 60    60 

I  Water,  distilled 77.6274 


60 


60 


Weight  of  a  Cubic  Foot  oj' 


Cast  iron 460.56 

Wrought  iron 486.65 

Steel 489.8 

Copper 565 

Lead 708.76 

Brass 537.75 

Tin 456 

White  pine... 29.56 

Loose  earth  or  sand 95 

Common  soil ; 124 

Strong  soil 127 

Clay 135 


lbs. 

Yellow  pine 33.81 

White  oak 35.2 

Live  oak , 70 

Salt  water  (sea) 64.3 

Freshwater 62.5 

Air 07629 

Steam 03689 

Clay 135 

Sand  113 

Cork  16 

TaJlow 59 

Brick 119 


MONEYS,   WEIGHTS    AND    MEASURES. 


MEASUEES. 

Linear  Measure  is  applied  in  measuring  length  and  dis- 
tance: 12  inches  =  1  foot,  3  feet  =  1  yard,  5^  yds.  =  1  rod, 
perch  or  pole,  40  rods  =  1  furlong,  8  furlongs  or  320  rods  --■- 
1  mile.  Sign-H,  in.  inches, /i5.  feet,  yd.  yard,  rd.  rod,  fur.  furlong, 
mi.  mile.     Furlongs  are  seldom  used.  5280  ft.  =  1  mile. 

1  palm  =  3  inches,  1  hand  =  4  inches,  1  span  =  9  inches,  1 
meter  =  3.28  feet. 

Scripture  Long  Measure. — A.  digit  =  .912  inches,  a  palm 
=  3.648  inches,  a  span  =  10.944,  a  cubit  =  1  foot  9.888  inches, 
Si  fathom  =  7  feet  3.552  inches. 

Jewish  Long  Measure. — A  cubit  =  1.824  feet,  a  Sabbath 
dojy^s  journey  =  3648  feet,  a  mile  =-.  7296  feet,  a  day^s  journey  = 
175104  feet,  or  33  miles  864  feet. 

Cloth  Measure. — Cloth  is  measured  by  the  yard  and  frac- 
tional parts  of  a  yard,  as  half,  quarter,  eighth,  sixteenth,  etc. 
The  yard  contains  3  feet,  or  36  inches. 

Marine  Measure. — Used  at  sea:  6  feet  =  1  fathom,  120 
fathonis  =  1  cable  length,  880  fathoms  =  1  mile. 

Metric  Long  Measure. — The  unit  of  Long  or  Lineal 
Measure  of  the  metric  system  is  tlie  Meter  (whence  the  name. 
Metric.)  The  Greek  prefixes,  deka,  etc. ;  and  the  Latin,  ded, 
etc.,  form  the  other  denominations  the  same  as  by  the  gram. 
The  meter  is  equal  to  39.3685  inches  of  our  linear  measure. 
The  signs  of  the  meter  and  denominations  above  are  written 
with  capitals :  M.  for  meter,  KM.  for  kilometer ;  those  of  th< 
denominations  below  the  meter,  with  small  letters — dm.,  deci- 
meter, etc. 

Surveyors'  Measure. — 7y%%  inches  —  1  link,  25  links  = 
1  rod  or  pole,  4  poles  or  100  links  =  1  chain,  80  chains  ^=  ] 
mile,  10  sq.  chains  =  1  acre,  640  acres  or  6400  sq.  chains  =  1 
sq.  mile  or  section  of  land. 

A  square  rod  contains  272^  sq.  feet.  An  acre  contains  4356C 
sq.  feet. 


MONEYS,   WEIGHTS    AND    MEASURES. 


Circular  Measure. — Uf^ed  in  reckoning  latitude,  longi- 
tude, etc.,  and  in  trigonometrical  calculation:  60  seconds  =  1 
minute,  60  minutes  =  1  degree,  30  degrees  =  1  sign,*  12  signs 
=  1  circle.  Signs. — ^^  seconds,  ^  minute,  °  degrees,  S.  sign,  C. 
circle. 

35°  3^  2^^  would  read  thirty-five  degrees,  three  minuteSj  and  two 


Measure  of  Time. — 60  seconds  =  1  minute,  60  minutes  = 
1  hour,  24  hours  =  1  day,  7  days  =  1  week,  30  days  =  1  lunar 
month,  365  days  =  1  year,  12  months  =  1  year. 

Square  Measure  is  used  for  measuring  surfaces:  144  sq. 
in.  =  1  square  foot,  9  square  feet  ^=  1  square  yard,  30J  square 
yards  =  1  square  rod,  160  square  rods  =  1  acre,  640  acres  =  1 
square  mile.     Signs,  sq.ft.,  sq.  yds.,  sq.  rds.,  A.,  M. 

Metric  Square  Measure. — The  unit  of  measure  for 
large  surfaces  is  the  Are,  from  whicli  are  derived  the  Hectare 
and  Centare.  For  smaller  surfaces  tlie  denominations  are  tlie 
same  as  for  measures  of  length,  with  the  addition  of  the  word 
square. 

Are  =  100  sq.  meters,  or  1  sq.  dekameter,  or  119.6  sq.  yds. 

Cubic  Measure  is  applied  to  solids,  and  comprises  length, 
breadth,  and  thickness,  or  depth.  A  cubic  foot  contains  1728 
inches,  that  is  12  times  12  times  12  inches;  a  cubic  yard  con- 
tains 27  feet,  or  3  times  3  times  3  feet. 

Metric  Cubic  Measure. — The  Stere  may  be  called  the 
imit  for  cubic  measure.  It  is  equal  to  a  cubic  meter  or  1.308 
yards. 

Wood  Measure. — Wood  is  sold  by  the  cord,  which  should 
contain  128  cubic  feet,  closely  piled,  and  138  feet  if  stowed  in 
a  boat  or  barge.  A  pile  of  wood  measuring  8  feet  long,  4  feet 
wide  and  4  feet  high,  contains  a  cord. 


10  MONEYS,    WEIGHTS   AND    MEASURES. 

Stone  Measure  is  used  for  measuring  masonry,  which  is 
sometimes  paid  for  by  the  foot,  but  usually  by  the  perch,  24| 
or  25  cubic  feet  1  perch;  the  former  for  private,  the  latter  for 
public  contracts,  as  railroad  or  government  work. 

A  wall  16J  feet  long,  IJ  feet  thick  and  1  foot  high  contains 
a  perch. 

Bricklayers'  Measure. — ^The  common  dimensions  of  a 
brick  are  8  inches  long,  4  inches  broad,  and  2  inches  thick. 
There  are  21  bricks  in  a  cubic  foot  of  wall,  including  mortar. 

A  wall  8  in.  or  1  brick  in  thickness  contains  14  bricks  to  the  sq.  ft.  of  surface. 
12      "      IJ  ''  "  "  21  ''  "  "  ** 

2g     a     2  "        "        "        28        "         **        "        " 

Dry  Measure.— Used  for  measuring  grain,  fruit,  etc.: 
8  quarts  =  1  peck,  4  pecks  =  1  bushel.  Signs,  qt.  quart,  'pk. 
peck,  hu,  bushel. 

Note.— The  bushel  is  a  cylindrical  vessel,  8  inches  deep  and  1834 
diameter,  inside,  and  contains  2150.42  cu.  in. 

Coal  Measure. — Coal  is  usually  sold  by  the  bushel,  which 
should  contain  2688  cubic  inches. 

Liquid  Measure. — For  measuring  all  liquids,  except  milk, 
beer  and  ale :  4  gills  =  1  pint,  2  pints  =  1  quart,  4  quarts  =1 
gallon.  Barrels,  tierces,  etc.,  are  no  longer  used  as  measures 
of  capacity ;  they  are  all  gauged  and  reckoned  by  gallons. 

Remark.— The  gallon  contains  231  cubic  inches. 

Ale  or  Beer  Measure. — The  gallon  contains  282  cubic 
inches,  and  the  number  of  pints  or  quarts  in  a  gallon  the  same 
as  in  Liquid  Measure. 

Metric  Measure  of  Capacity. — The  lAter  is  the  unit 
of  measure  for  capacity,  and  is  equal  to  a  cubic  decimeter  or 
1.0567  quarts  of  United  States  liquid  measure. 


MISCELLANEOUS. 


11 


MISCELLANEOUS. 

Effects  of  CoaL — Small  coal  produces  about  f  the  effect 
of  large  coal  of  the  same  species. 

CharcoaL — The  best  quality  is  made  from  oak,  maple, 
beecli  and  chestnut.  Wood  will  furnish,  when  properly  burnt, 
about  16  per  cent,  of  charcoal.  A  bushel  of  charcoal  from 
hard  wood  weighs  about  30,  from  pine,  about  29  lbs. 

Coke. — A  bushel  of  the  best  coke  weighs  32  lbs.  Coal  fur- 
nishes from  60  to  70  per  cent,  of  coke  by  weight. 


Kelative  Heating  Power  of  Different  Kinds  op 
Fuel,  according  to  Weight. 

Charcoal  100,  mineral  or  stone  coal  82^,  dry  wood  48 J. 
Hence,  if  a  tun  of  charcoal  cost  no  more  than  a  tun  of  mineral 
coal,  the  former  would  be  the  cheaper  fuel  by  21  per  cent. 

Kelative  Heating  Power  of  Different  Kinds  op 
Wood,  according  to  Measure. 


Shell-bark  Hickory 
Pignut,      .     . 
White  Oak,  . 
White  Ash,  . 
Dog  Wood,    . 
Scrub  Oak,    . 
Witch  Hazel, 
Appletree, 
Red  Oak,  .     . 
White  Beech, 
Black  Walnut, 
Black  Birch, 


100 
93 
81 
77 
78 
73 
72 
70 
69 
65 
65 
63 


Yellow  Oak, 
Hard  Maple, 
White  Elm,  . 
Red  C'edar,  . 
Wild  Cherry, 
Yellow  Pine, 
Soft  Maple,  . 
Chestnut,  .  . 
Yellow  Pophir, 
Butternut,  . 
White  Birch, 
White  Pine,  . 


60 
60 

58 
56 
55 
54 
54 
52 
52 
51 
48 
42 


Digging. — 23  cubic  feet  of  sand,  or  18  cubic  feet  of  earth, 
or  17  cubic  feet  of  clay  make  a  tun.  18  cubic  feet  of  gravel, 
or  earth,  before  digging,  make  27  cubic  feet  when  dug. 


12  '       MISCELLANEOUS. 


Gas. — 1.43  cubic  feet  of  gas  per  hour  give  a  light 
equal  to  that  of  a  candle;  1.96  cubic  feet  equal  4 
candles  ;  3  cubic  feet  equal  10  candles. 

Horse  Power  in  machinery  is  reckoned  at  33,000 
lbs.  raised  one  foot  in  a  minute,  but  the  ordinary 
work  of  a  horse  is  only  22,500  lbs.  per  minute  for 
8  ho^rs. 

Strength  of  a  Man. — The  mean  effect  of  the 
power  of  a  man,  unaided  by  a  machine,  is  the  rais- 
ing 70  lbs.  1  foot  high  in  a  second  for  10  hours  a 
day=i  of  the  power  of  the  horse. 

Note. — Two  men  working  at  a  windlass  at  right  angles  to  each 
other,  can  raise  70  lbs.  more  easily  than  one  man  can  30  lbs. 

A  foot  soldier  travels  70  yards,  making  90  steps 
in  one  minute,  common  time. 

In  quick  time,  86  yards,  making  110  steps. 
In  double  quick,  109  yards,  making  140  steps. 
Average  weight  of  men,  150  lbs.  each. 
Five  men  can  stand  in  a  s]Dace  of  1  square  yard. 

A  man  without  a  load  travels  on  a  level  ground 
8-|  hours  a  day,  at  the  rate  of  3.7  miles  an  hour,  or 
31J  miles  a  day.  He  can  carry  111  lbs.  11  miles  in  a 
day. 

A  porter  going  short  distances  and  returning  un- 
loaded, can  carry  135  lbs.  7  miles  a  day.  He  can 
carry  in  a  wheelbarrow  150  lbs.  10  miles  a  day. 

The  muscles  of  the  human  jaw  exert  a  force  of 
534  lbs. 

Hay. — 10  cubic  yards  of  meadow  hay  weigh  a 
tun.  When  the  hay  is  taken  out  of  old,  or  the  lower 
part  of  large  stacks,  8  to  9  cubic  yards  will  make  a 
tun.  10  to  12  cubic  yards  of  clover,  when  dry,  weigh 
a  tun. 

Hills  in  an  Acre. — 3  feet  apart,  there  are  4840 
hills  in  an  acre. 


PAPER — SUNDRIES. 


13 


PAPER. 


SIZES  OF  PAPER  MADE  BY  MACHINERY, 


FLAT  PAPER. 


Letter, 10X16 

Com'l  Letter,     .    .    .  11X17 

Packet, 12X19 

Foolscap, 13X16 

Cap,         14X17 

Crown, 15X19 

Demy, 16X^1 

Folio,       17X22 

Check  Folio,  ....  17X^4 


Tax  Duplicate, 
Medium, 
Royal,      .    . 
Super  Royal, 
Elephant,    . 
Imperial, 
Columbier, 
Atlas,  .    .    . 
Antiquarian, 


.  17X30 
.  18X23 
.  19X24 
.  20X28 
.  23X28 
.  23X31 
.  23X34 
.  26X33 
.  31X53 


FOLDED  PAPER. 


DESIGNATIONS  OF  SHEETS  ACCORDING  TO  FOLDS  OF  PAPER. 

Folio. — A  sheet  folded  in  two  leaves. 
Quarto. — A  sheet  folded  in  four  leaves. 
Octavo. — Or   8vo,  a  sheet  folded  in  eight  leaves. 
Duodecimo. — Or  12mo*,  a  sheet  folded  in  twelve  leaves. 

24  sheets  =  1  quire,  20  quires  =  1  ream,  2  reams  =  1  bundle. 

Book-binders  count  from  16  to  20  sheets  to  a  quire  in  binding 
account  books. 

WRAPPING  PAPER. 

Wrapping  paper  is  sold  by  the  bundle,  which  are  generally 
^hort  count.  The  full  cou7it  reams  contain  20  qrs.  of  24  sheets 
each. 


SUNDRIES. 


12  articles  =  1  dozen. 
12  dozen  =  1  gross. 
12  gross  =  1  great  gross. 
20  articles  =  1  score. 
1  barrel  =  200  lbs. 


14  lbs.  of  flour  =  1  stone. 
14  stones  of  flour  =  1  bbl. 
1  bbl.  of  flour  =196  lbs. 
1  barrel  =  31 M  gallons. 
1  hogshead  =  2  bbls. 


♦  The  size  of  this  book  is  12mo. 


14  THERMOMETERS. 


THERMOMETERS. 

The  Celsius  or  Centigrade  thermometer  has  the  zero  at 
the  freezing  point  of  water,  and  the  distance  between 
that  and  the  boiling  point  of  water  divided  into  100 
degrees, — hence  the  name  Centigrade. 

The  Beaumur  thermometer  has  the  zero  at  the 
freezing  point,  and  80°  between  that  and  the  boiling 
point  of  water. 

The  Fahrenheit  thermometer  has  the  zero  at  32° 
below  the  freezing  point  of  water,  and  has  180° 
between  freezing  and  boiling  point  of  water. 

To  convert  degrees  of  Centigrade  into  degrees  of 
Fahrenheit,  multiply  the  degrees  of  Centigrade  loy 
9,  divide  the  product  by  5,  and  add  32  to  the  quo- 
tient, the  answer  will  be  degrees  of  Fahrenheit. 

To  convert  degrees  of  Eeaumur  into  degrees  of 
Fahrenheit,  multiply  the  degrees  of  Eeaumur  by  9, 
divide  the  product  by  4,  and  add  32  to  the  quotient, 
the  answer  will  be  degrees  of  Fahrenheit. 

To  convert  degrees  of  Fahrenheit  into  Centigrade, 
subtract  32  from  the  degrees  of  Fahrenheit,  multiply 
the  remainder  by- 5,  and  divide  the  j^roduct  by  9. 

To  convert  degrees  of  Fahrenheit  into  Eeaumur, 
subtract  32  from  the  degrees  of  Fahrenheit,  multipl}^ 
the  remainder  by  4,  and  divide  the  product  by  9. 

To  convert  Centigrade  into  Eeaumur,  multiply  the 
degrees  of  Centigrade  by  4  and  divide  the  product 
by  5, 

To  convert  Reaumur  into  Centigrade,  multiply  the 
degrees  of  Eeaumur  by  5  and  divide  the  product 
by  4. 

The  sum  of  the  degrees  of  Centigrade  and  Eeau- 
mur plus  32  will  give  the  degrees  of  Fahrenheit. 


THE  CONCISE 

MEROAKTILE  ARITHMETIC. 


I.  INTRODUCTION. 

AEITHMETICAL  DEFINITIONS. 

Article  1.    Arithmetic  is  the  science  of  numbers. 

Art.  2.  The  theory  of  Arithmetic  treats  of  the 
properties  and  relations  of  numbers. 

Art.  3.  The  practice  of  Arithmetic  shows  the 
application  -of  number  to  business,  the  mechanics' 
art,  etc. 

Art.  4.  Quantity  is  any  thing  that  can  be  in- 
creased or  diminished. 

Art.  5.  Notation  is  the  art  of  representing  num- 
bers by  figures. 

Art.  6.  Numeration  is  the  art  of  reading  figures 
when  arranged  to  represent  numbers. 

Art.  .7.  The  four  fundamental  rules  of  Arithme- 
tic are :  Addition,  Subtraction,  Multiplication,  and 
Division. 

Art.  8.  Addition  is  the  art  of  uniting  two  or 
more  numbers  into  one.  The  result  obtained  by 
adding  is  called  Amount  or  Sum. 


16  ARITHMETICAL   DEFINITIOJ^S. 

Art.  9.  Subtraction  is  the  method  of  finding  the 
difference  between  two  numbers.  The  result  ob- 
tained is  called,  Remainder, 

Art.  10.  Multiplication  is  the  process  of  taking 
one  number  as  many  times  as  there  are  units  in 
another.     The  result  obtained  is  called,  Product. 

Art.  11.  Division  is  the  method  of  ascertaining 
how  many  times  a  given  number  is  contained  in 
another.     The  result  obtained  is  called,  Quotient. 

Art.  12.  Percentage  is  the  method  of  reckoning 
by  hundredths. 

Art.  13.  A  Fraction  is  a  part  or  a  number  of 
parts  of  a  whole. 

Art.  14.  Interest  is  a  percentage  allowed  for  the 
use  of  capital. 

Art.  15.  Bank  Discount  is  a  percentage  deducted 
from  capital  loaned  for  the  use  of  such  capital. 

Art.  16.  True  Discount  is  the  difference  between 
the  present  worth  of  a  note  and  the  amount  for  which 
it  is  drawn. 

Art.  17.  Proportion  is  an  expression  of  equal 
ratios. 

Art.  18.  Arithmetical  Signs. 


+ 
X 


8  the  sign  of  equality. 
8  the  sign  of  addition. 
8  the  sign  of  subtraction. 
s  the  sign  of  multiplication. 
8  the  sign  of  division. 
s  the  decimal  sign. 
s  the  sign  of  proportion. 
s  the  sign  of  percentage. 


NOTATION    AND    NUMERATION.  17 


11.  NOTATION  AND  NUMERATION. 

Article  1.  Notation  is  the  art  of  representing 
numbers  by  symbols,  called  figures  or  digits.  There 
are  ten  of  these  figures : 

0123456789 

nought    one       two     three     four      five        six      seven   eight     nine 

The  first  is  also  called  zero^  or  cipher. 

Art.  2.  When  a  larger  number  than  nine  is  to  be 
represented,  two  or  more  figures  are  used. 

Art.  3.  Numeration  is  the  method  of  reading 
these  figures  when  arranged  to  represent  numbers. 
Eor  this  purpose  they  are  usually  divided  into  pe- 
riods from  the  right, 

COMMON  METHOD. 

Art.  4.  According  to  the  Common  or  French  method 
of  numeration,  the  first  period  on  the  right  contains 
units,  tens,  and  hundreds. 

12        3. 

hundreds  tens      units. 

The  second  period  contains  units,  tens  and  hundreds 
0^  thousands ;  the  third,  units,  tens  and  hundreds  of 
millions;  the  fourth,  billions;  the  fifth,  trillions;  the 
sixth,  quadrillions;  the  seventh,  quintillions ;  the 
eighth,  sextillions ;  the  ninth,  septillions ;  the  tenth, 
octillions;  the  eleventh,  nonillions ;  the  twelfth,  de- 
cillions. 

The  higher  denominations  are  formed  by  prefix- 
ing to  decillions  the  Latin  words,  uno,  duo,  tre,  qua- 
tuor,  quin,  sex,  septen,  octo,  noven. 

ENGLISH  METHOD. 

Art.  5.  According  to  the  English  method,  the  first 
six  orders  have  the  same  names  and  signification  as 
2 


18  NOTATION    AND    NUMERATION. 

those  of  the  French.  Every  period,  however,  con- 
sists of  six  orders.  The  second  period  is  million; 
the  higher  denominations  are  named  the  same  as 
by  the  Common  method,  but  have  different  signifi- 
cations. 

Remark. — It  will  be  noticed  that  each  period,  according  to 
tlie  Common  method^  is  one  thousand  times  the  preceding  one, 
and  according  to  the  English  method  one  million  times.  Hence, 
according  to  the  Common  method,  a  billion  is  a  thousand 
millions,  and  according  to  the  English  a  billion  is  a  million 
millions. 

26,839,506,720,052,005  according  to  the  Common 
method  would  read :  Twenty-six  quadrillions,  eight 
hundred  and  thirty-nine  trillions,  five  hundred  and 
six  billions,  seven  hundred  and  twenty  millions, 
fifty-two  thousand  and  five. 

The  same,  according  to  the  English  method, 
would  be  pointed  off  thus:  26839,506720,052005, 
and  read,  twenty-six  thousand  eight  hundred  and 
thirty-nine  billions,  f^YQ  hundred  and  six  thousand 
seven  hundred  and  twenty-millions,  fifty-two  thou- 
sand and  five. 

EOMAISr  NOTATIOISr. 

Art.  6.  In  Roman  notation  numbers  are  repre- 
sented by  letters,  as  follows :  I,  one ;  V,  five ;  X, 
ten;  L,  fifty;  C,  one  hundred;  D,  five  hundred; 
M,  one  thousand.  A  line  over  a  letter  increases  its 
value  one  thousand  times:  thus,  I>  denotes  ^yq 
hundred  thousand.  A  letter  of  less  value  placed 
before  one  of  greater  value  diminished  the  latter 
the  amount  of  the  value  of  the  former:  thus,  -CM 
denotes  nine  hundred. 

MCMLYDXEYII  reads,  one  million  nine  hun- 
dred fiiPty-five  thousand,  five  hundred  and  forty- 
seven.     MDCCCLXXYI  =  1876. 


ADDITION.  19 


III.  ADDITION. 

Art.  1.  The  process  of  uniting  two  or  more  num- 
bers into  one  is  called  Addition. 

Art.  2.  The  result  obtained  is  called  sum^  amount ^ 
totals  OY  footing. 

Art.  3.     The  sign  +,  when  placed  between  two 

numbers,  indicates  that  they  are  to  be  added  together. 

Note. — This  book  being  a  Mercantile  Arithmetic,  it  is 
thought  best  to  omit  short  examples  in  addition,  as  the  parties 
using  the  same  are  supposed  to  be  acquainted  with  the  funda- 
mental rules  of  the  science,  but  need  to  acquire  accuracy  and 
rapidity.  The  principles  will,  therefore,  be  stated,  and  such 
hints  given,  which,  when  put  into  practice,  will  enable  the 
learner  to  add  up  rapidly  and  correctly. 

Art.  4.  It  is  necessary,  in  performing  the  opera- 
tions in  addition,  to  place  units  under  units,  hundreds 
under  hundreds,  etc. 

EXAMPLES. 

To  add  forty,  three  hundred  and  seventy-two,  one 
thousand  eight  hundred  and  sixty-seven,  and  eight 
hundred  and  ninety-five,  they  should  be  arranged  as 
follows  : 

40 

372 

1867 

895 

3174   ^. 

We  commence  the  process  by  Adding  the  right  hand  or  unit 
column,  beginning  with  the  lower  figure,  thus :  5  and  7  are  12, 
and  2  are  14,  that  being  4  units  and  1  teen. 

The  unit  (4)  is  placed  under  the  unit  column  as  the  result, 
and  the  teen  (1)  is  added  to  the  second  or  teens  column. 

Next  the  teens  are  to  be  added,  thus :  1  (the  1  tee q- obtained 
by  adding  the  unit  column)  and  9  are  10,  and  6  are  16,  and  7 
are  23,  and  4  are  27;  namely,  27  teens  or  7  teens  and  2  hundred. 


20 


ADDITION. 


The  7  teens  are  placed  under  the  teens  column  as  the  result, 
and  the  2  hundred  are  added  to  the  third  or  hundreds  column, 
and  the  hundr-ed  column  is  added  in  the  same  way,  resulting  in 
21  as  the  answer;  namely,  21  hundred,  or  one  hundred  and  2 
th(msand.  The  one  is  placed  under  the  hujidred  column  as  the 
result,  and  2  thousand  is-added  to  the  1  thousand  in  tlie  ex- 
ample, resulting  in  three  thousand,  which  3  is  placed  under  the 
thousand  column  as  the  answer — the  whole  footing  will  now 
read  (commencing  at  the  left)  three  thousand  one  hundred  and 
seventy-four  (3174). 


Add  the  following: 


2365 

92245 

925683 

7629548 

18293 

28392 

968542 

9832965 

8769 

67268 

768656 

7629824 

2965 

63629 

329871^ 

4567897,^ 

3276C 

24432^ 

123456 

7632851 

Art.  5.  In  order  to  acquire  rapidity,  the  learner 
should,  from  the  beginning,  avoid  counting  by  their 
fingers,  but  should  familiarize  themselves  with  the 
catch  figures.  The  catch  figure  is  the  unit  figure  of 
the  result  of  adding  two  units  together;  thus:  When- 
ever 5  and  6  are  added  together,  the  unit  figure  in 
the  result  (11)  is  one ;  whenever  6  and  9  are  com- 
bined, the  unit  figure  is  5. 

Class  exercises  on  giving  the  catch  figure  and  on  applying  it 
will  be  found  both  interesting  and  profitable. 

The  exercise  may  be  conducted  as  follows: 

Where  7  and  9  are  combined,  the  unit  figure  is — ?  (6).  17 
and  9  are  —  ?  (26).  37  and  9  —  ?  67  and  9  —  ?  27  and  9  —  ? 
.The  teacher  putting  the  questions,  and  the  scholars  in  concert 
Responding  by  giving  the  answer. 

5.  13965+6725+68349+76587+9825+99542=  * 

^  6.  2592+18596+9382+6732+95876+29326= 

7.  8549+8329+6784+7376+92542+93586= 

,    8.  3576+7654+3295+7628+27654+7629= 

9.  3733+9258+8975+9268+9327+7652= 

10.  6686+8259+9762+3876+8585+7895= 

11.  2936+9286+7654+6832+9257+6873= 

*  The  answers  will  be  found  at  the  close  of  the  chapter 
(page  24). 


ADDITION.  21 


Art.  6.  The  process  of  adding  Federal  Money 
differs  from  the  foregoing  only  in  the  use  of  the 
dollar  ($ )  and  decimal  (  . )  signs. 

12.  $568.32+,  $965+,  $985.20^<^  $— ? 

Note. — In  adding  federal  money  the  learner  must  "be  care- 
ful to  place  the  decimal  points  (the  sign  separating  the  dollars 
and  cents)  of  the  amounts  of  money  to  be  added  under  each 
other,  thus: 

$568.32 
965. 
985.20 

13.  $375.15+  $950.+  $876.51+  $7.57+  $987.56+ 
$781.+  $659.16+  $286.56+.56+  $185.20r= 

/    14.  $878.10+  $758.+  $238.68+  $875.+  $658.99+ 
/$878.+  $751.87+  $2.85+  $286+289.54= 

15.  $751.+  $518.91+  $361.98+  $678.10+  $777.67 
+  $765.+  $958.+  $392.51+  $682.19+  $775.20= 

16.  $868.19+  $18.+  $85.88+  $567.50+  $678.96+ 
$879.+  $759.15+  $894.26+  $824.18+  $982.56= 

17.  $781.59+  $759.10+  $899.99+  $569.+  $569.78 
+  $656.71+  $871.+  $326.50+  $98.27+  $976.58= 

18.  $798.15+  $7.76+  $786.56+  $437.+  $788.15+ 
$788.88+  $935.62+  $92.52+  $768.92= 

19.  $889.+  $878.99+  $878.95+  $898.10+  $897.+ 
$987.54+  $651.25+  $329.77+  $628.95+  $628.92= 

20.  $18146+$71.25+$641.04+$4501+$87700= 

21.  $1770.03+$1006.01+     $364.01+$5442.99= 

22.  $2310.00+$1068.24+$26107.18+$2136.18= 

23.  $109.79+  $999.99+     $666.56+  $449.99= 

24.  $777.00+$7999.00+  $6666.00+$6730.15= 

25.  A  merchant  has  29  pieces  of  silk  in  1  package, 
35  in  another,  79  in  a  third.  In  the  first,  there  are 
1497  yards,  in  the  second,  2173,  in  the  third,  4130. 
How  many  pieces,  and  how  many  yards  in  all? 

26.  $1.23+283+$685.04+$123.45+$78= 


22  ADDITION. 


27.  31465+2316532+107+3790+465321+36545G3+ 
107653+23650+1007+30o72+503102+21063  is  how 
much  ? 

28.18230+476+41034+9875+65432+5678+12090+ 
9387+8276+565 +  13654+443z=z:how  much? 

Ans.  Sum  of  27  and  28,  73440G5. 
.J^J)  29.  46853  +  9654+45679  +  9837+18708+7967+485 
>yS^78963+84989+12345+7069+8090+7483+96748==? 

TAKING  TWO   AND  THREE   FIGURES    AT   A  TIME. 

To  enable  scholars  to  grasp  two  and  three  figures  at  a  time,  and 
carry  them  up  as  one,  they  might  be  exercised  on  the  blackboard 
in  such  sums  as  the  following: 

136377436      146739213698 
954186987      782163846673 

2  1  3  4  1  4  1  3  6  .2  1  2 

Such  exercises  ought  to  be  of  frequent  occurrence  and  scholars 
encouraged  to  answer  in  concert. 

The  answers  should  be  given  instantaneously,  naming  only  the 
unit  figure,  as  shown  in  the  column  below: 

8456  I  ^  After    writing  on   the  right  of  the  first  column  the 

1345  /  figures  produced  by  pairing,  the  teacher  may  lead  the 

156^1^  class  in  adding,  thus:   17  and  3?     30  and  1?     41  and 

9456  >  ^^     47  and  7?     54  and  1?     65  and  6?     81  and  6?     96 


}5      and  2?     108  and  11? 


8998  \n  It  will  be  observed  that  the  tens  produced  in  forming 

1898  j  the  pairs  were  not  named.     The  same  course  should  be 

1 A7Q  r  ^  pursued  in  the  class,  as  the  learner  is  unconscious  of 

1684  1  making  as  great  an  effort  as  he  really  does. 

7893  )  '  When  the  ten  is  omitted  by  mistake,  attention  should 

1453  )p  be  called  to  it  by  giving  the  full  number,  as  15  or  11 


;} 


1763  /  instead  of  5  or  1. 

Qft7A  \  ^  "^^^  other  columns  should  be  added  without  the  aid 

7897  ^  °^  ^^^  marginal  figures. 

2586  j  ^  After    thorough    drill    in    this,    the   class   should    be 

8529^-  taught  to  take  three  figures  and  feven  four  as   rapidly 

1438  /  as  one. 


ADDITION.  23 


30.  Find  the  sum  of  8934,  16749,  809,  67549,  98697, 
746839,  1498,  829555,  9218967,  8347912,  968000,  74685. 
Total  of  the  preceding  two,  20815046. 

Foot  up  the  following  columns : 


31 

32 

33 

34 

35 

31645 

3454 

4213 

1565 

3654 

98760 

2136 

6314 

3657 

1095 

3G875 

1364 

2316 

5437 

9014 

57893 

4633 

1369 

3457 

6789 

14567 

9897 

9306 

1234 

9687 

34564 

7879 

6039 

3421 

5764 

46387 

2164 

8109 

6789 

1567 

93178 

4163 

9876 

1746 

9139 

78163 

4569 

6789 

3456 

1456 

64518 

5496 

4567 

1378 

2345 

17514 

6428 

5679 

5932 

5432 

45678 

8297 

3263 

4567 

6542 

21364 

9287 

9457 

1657 

1395 

7198 

7928 

1459 

6574 

3642 

3165 

9872 

1455 

5638 

1365 

4124 

8729 

•9375 

4932 

2315 

1345 

9314 

5976 

1397 

9365 

3146 

3162 

7639 

9765 

3510 

4165 

2136 

7938 

3765 

1096 

3216 

9364 

3959 

1456 

3765 

36.  Add  together  the  following  numbers:  313,  2109, 
6785,  2736,  798,  987,  21363,  316,  4934,  2178,  1009,  396, 
298,  2753,  607,  3145,  213,  6709,  6093,  190,  2130,  2160, 
716,  213,  9876,  45678,  2137,  2198,  9039,  6789,  3097, 
4684,  2136,  2178,  5672,  1987,  6789. 

Answers  promiscuously  arranged:  95368,  77823, 120272, 
115098,  667465,  88937,  171411. 

The  Teacher  should  not  permit  his  scholars  to  divide  these  col- 
umns when  adding,  nor  should  he  allow  them  to  resort  to  the  aid 
of  strokes  or  practice  counting  on  their  fingers. 


37 

38 

39 

40 

41 

3286 

2467 

34564 

46321 

3614 

6713 

109 

12345 

13632 

1364 

3654 

3178 

65435 

14567 

5436 

176 

145 

87654 

53678 

7835 

3976 

6178 

34564 

86367 

4678 

6345 

4156 

13682 

85432 

8793 

9823 

7532 

75671 

36457 

701 

6023 

9890 

86317 

21836 

9804 

1367 

6821 

24328 

17354 

1306 

8965 

9854 

98713 

63542 

717 

8632 

3821 

21345 

78163 

2103 

1034 

5843 

1286 

82645 

6397 

6312 

1936 

78654 

34685 

1096 

4593 

7136 

19876 

31768 

2130 

3687 

9876 

93643 

65314 

3107 

5006 

2863 

6356 

68231 

167 

7164 

123 

78397 

64037 

2109 

1763 

7436 

21602 

34685 

3678 

2139 

1567 

71346 

35962 

2176 

8236 

2563 

28653 

21363 

5432 

7860 

8432 

17648 

78636 

2137 

3613 

1345 

82351 

19854 

28639 

109 

8736 

21368 

80145 

1765 

1756 

8654 

78631 

87654 

371 

6386 

1263 

17639 

12345 

71031 

9890 

1345 

82360 

78654 

1463 

8243 

3093 

45671 

12345 

3168 

Answers:  42838.  48213,  217166,  274993,  162504, 
45063,  37293080,  25668,  275966,  3116208,  185140, 
378363,  434870,  136751,  126362,  1300099,  181217, 
171411,  88937,  77823,  115098,  120272,  66746,  20380194, 
57436,  7158925,  1325672,  $2518.52,  $5617.03,  $6557.68, 
$6660.56,  $5109.27,  $6508.52,  $5403.86,  $7668.47,  $31- 
621.60,  $2226.33,  $43.00,  $22172.15,  7800,  2500,  2450, 
21621200,  1243883. 


SUBTRACTION.  25 


IV.  SUBTRACTION. 

Art.  1.  The  process  of  taking  a  lesser  number  or 
quantity  from  a  greater  of  the  same  kind  or  denom- 
ination is  called  Subtraction. 

Art.  2.  The  result  obtained  is  called  difference^  re- 
mainder, or  excess. 

Art.  3.  The  sign  of  subtraction  is  — ,  and  is  called 
minus.     8  —  2  reads  eight  minus  two. 

Art.  4.    1.  Find  the  difference  between  786  and  323. 

Solution. — We  place  the  smaller  number  under  the 
larger  one,  units  under  units  and  hundreds  under  hun-      ^86 
dreds,  etc.,  and  proceed  to  subtract  from  the  right  to  the 
left,  viz :    3  from  6  leaves  3  ;  next  we  subtract  the  teens : 
2  from  8  leaves  6,  this  we  place  in  the  teens  place ,   3 


323 

^  ii'Uiii   o   leiives   u,    liii»    we   pjaue   in    Lue    Lceii.-s    pju-Lc  ,    o        AQO 

(hundred)  from  7  (hundred)  leaves  4  (hundred),  which      ^"*^ 
is  placed  in  the  hundreds  place.     The  answer  (Remain- 
der) is  463, 

2.  23964—  12853  3.  2986—  258 
4.  6972325—  4232323  5.  6896542—  84312 
6.  276289995  —  16278585   7.  32987632  —  11976412 

Art.  5.  1.  Find  the  difference  between  5354  and 
897. 

Solution. — In  this  case  we  find  that  the  several  j  ^^  JJ' ^^ 

numbers  in  the  5it6^?-a/m7i(i*  are  greater  than  those  of  2*3'5*4 

the  minuend.^    We  can  not  take  7  from  4,  so  we  take  o  q  ^• 

one  from  the  teens  in  the  7mn.  and  add  it  to  the  units,  ^  9  ' 


making  14  —  7  from  14  leaves  7.  Having  taken  one  1  4  5  7 
from  the  teens,  leaves  4.  9  from  four  we  can  not 
take,  so  we  again  take  one  of  the  next  figure  in  the 
min.  (3)  and  add  it  to  the  teens,  making  again  14.  9  from  14 
leaves  5.  8  from  2  (1  having  been  taken  from  the  3  to  add  to  the 
teens)  we  can  not  take,  so  we  proceed  to  take  1  from  the  thou- 
sands, which  makes  10  (hundred)  +2  (hundred)  gives  us  12  less 
8  makes  4.  One  having  been  taken  from  the  2  (thousand)  leaves 
1.     Answer,  1457. 

^Subtrahend,  the  number  to  be  taken  from  the  minuend. 
fMinuend,  the  greater  number,  from  which  the  lesser  is  to  be 
subtracted. 


2G 


SUBTRACTION. 


2.  2345678—  689829 
4.  6123546  —  5261862 
6.     3254298  —  1185169 


3.  621129  —  509826 
5.  921654  —  629827 
7.     325627  —  124939 


Art.  6.  In  order  to  ascertain  the  difference  be- 
tween the  sum  of  two  columns,  subtraction  may  be 
formally  dispensed  with  by  adding  the  largest  column 
first,  and  by  adding  in  the  difference  thus : 


$286.98 
385.46 
928.54 
326.28 


Solution.— Having  obtained  the 
sum  of  the  larger  column,  $1927.26, 
we  proceed  to  add  up  the  smaller, 
viz:  2  -I-  6  +  2  are  9  and  7  (to  make 
the  result  ^he  same  as  the  units 

in  the  larger  column)  are  16 

1  (carried  from  the  result  of  the 
units)  +  4  +  3  +  4  =  12,  the  unit  fig- 
ure of  this  result  being  the  same 
as  that  of  the  dimes  in  the  larger 

column,  O  is  the  difference 1  (carried  from  the  dimes  column) 

+  64-9  +  5  =  21 +  6(  to  make  the  difference)  =  27 2  +  6  +7  +  8  =» 

23;  -r  9  (to  make  the  difference)  =  32 3+7+2  +  3  =  15  +  4  to 

make  the  difference  =  19.     Thus  giving  as  the  difference,  $496.07. 


.  $385.42 
279.35 
766.42 

$1927.26 


$1927.26 


Proof: 


S385.42 
279.35 
766.42 

$1431.19 


$1927.26  sum  of  large  column. 
$1431.19  sum  of  small  column. 

$496.07  difference. 


Find  the  difference,  by  addition,  of  the  following: 


2. 


$628.93 
542.69 
392.75 
826.37 
978.62 
126.58 


$258.72 
385.98 
726.18 
195.42 
329.54 


3. 


$3852.19 
6829.16 
9325.18 
2762.29 
3218.75 


$625.28 
398.75 
285.32 
975.68 
932.85 


$3495.94 

Ans.:  463,  11111,  2728,  2740002,  6812230,  2600- 
11410,  21011220,  1457,  $496.07,  $1600.10,  $22769.69, 
101303,  $25170.26,  $17796.82,  200688,  2069129,  291- 
827,  1655849,  861684. 


'•■•This  method  was  suggested  to  the  author  by  E.  P.  Goodiiough,  Esq. 


THE    COMPLEMENT  27 


THE   COMPLEMENT. 

Taking  the  Complement,  or  "  making  change," 
is  the  process  of  subtracting  a  lesser  number  from 
a  ''round  sum."  It  is  emploj^ed,  as  the  second 
term  indicates,  in  making  change  or  finding  the 
sum  to  be  paid  back  to  the  payer  out  of  the  amount 
handed  by  him  in  pa^^ment.  The  complete  number 
is  always  the  sum  of  one  or  more  of  the  denomina- 
tions of  coin  or  currency — $1,  $5,  $10,  50c.,  25c.,  etc. 
It  will  be  found  that  the  complement  of  the  teens  is 
always  in  the  SOs,  the  complete  number  being  $1 ; 
the  payment  to  be  made,  lie. — complement,  89; 
the  payment  to  be  19c. — complement,  81.  The 
complement  of  the  20s  in  the  70s ;  that  of  the  308 
in  the  60s;  of  the  40s  in  the  50s;  of  the  50s  in  the 
408;  of  the  60s  in  the  308;  of  the  708  in  the  20s, 
etc. 

It  will  be  found  to  be  a  very  profitable  class-drill,  to  conduct 
an  exercise  on  making  change  in  the  following  way : 

Teacher.  The  complete  number  being  $3,  what  is  the  com- 
plement out  of  a  payment  of  $1.50?  (The  class  calls  out  the 
complement,  $1.50.)  The  drill  is  conducted  with  enthusiasm 
for  some  time  on  the  same  complete  number  without  naming 
it  again,  naming  a  different  payment,  thus:  The  complete 
number  being  $5,  payment  $3.25,  complement — ?  pavment 
$1.85?  $1.75?  $3.55?  $4.50?  50c.?  75c.?  85c.?  $3.60, 
etc. 

The  students  should  be  required  to  give  the  denomination 
of  the  answer,  whether  in  dollars,  cents,  etc.  In  a  short  time 
the  students  will  find  it  an  advantage  to  subtract  from  the  left 
to  the  right  instead  of  the  reverse,  by  taking  the  $,  calling 
$5.00  $4,910.  We  do  not  think  it  advisable  to  require  the 
student  to  thus  subtract  from  the  left,  but  his  attention  may 
be  called  to  the  practicability,  and  if  he  find  it  of  advantage, 
he  should  use  it  If  the  habit  is  once  acquired,  it  will  facili- 
tate the  taking  of  the  complement  materially.  We  have  con- 
ducted a  class  exercise  in  schools  where  it  had  never  been 
taught,  and  in  the  -course  of  a  half  hour  the  complement  was 
given  by  the  entire  class  insianter. 


28 


MULTIPLICATION. 


V.  MULTIPLICATION. 

Art.  1.  Multiplication  is  a  short  method  of 
adding.  X  is  the  sign.  3x6  =  18,  reads,  three  times 
six  equals  eighteen. 

MULTIPLICATION  TABLE. 


IX  1- 

1 

2X    1  = 

2 

3X    1=      3 

4X    1=     4 

IX  ^  = 

2 

2X    2- 

4 

3X    2=:.      6 

4X    ^=      8 

IX    3=3 

3 

2X    3  = 

6 

3X    3=      9 

4X    3=r    12 

IX    4  = 

4 

2X    4  = 

8 

3X    4^    12 

4X    4=    16 

IX   5  = 

5 

2X    5  = 

10 

3y    5=    15 

4X    5=    20 

IX    6  = 

6 

2X    6:.:. 

12 

3X    6==.    18 

4X    6=    24 

IX    7^ 

7 

2X    7... 

14 

3X    7=    21 

4X    7=    28 

IX      8:= 

8 

2X    8=. 

16 

3X    8=    24 

4X    8=    32 

IX-  9  = 

9 

2X    9:- 

18 

3X    9=    27 

4X    9=    36 

1X10  = 

10 

2X10  = 

20 

3X10=    30 

4X10=    40 

1X11- 

11 

2X11  = 

22 

3X11=    33 

4X11=   44 

1X12  = 

12 

2X12  = 

24 

3X12=    36 

4X12=    48 

6X    1  = 

5 

6X    1  = 

6 

7X    1=      7 

8X    1=     8 

5X    2  = 

10 

6X     2:z= 

12 

7X    2==:=    14 

8X    2r=    16 

6X    3  = 

15 

6X    3  = 

18 

7X    3=   21 

8X    3=   24 

5X    4  = 

20 

6X    4  = 

24 

7X     4r=:.    28 

8X    4=    32 

5X    5  = 

25 

6X    5  = 

30 

7X    5=    35 

8X    5=    40 

6X    6  = 

30 

6X    6  = 

36 

7X    6=    42 

8X     6=:=:    48 

6X    7  = 

35 

6X    7  = 

42 

7X    7r=   49 

8X    7==    56 

5X    8  = 

40 

6X    8  = 

48 

7X      8r:r:     56 

8X    8=    64 

5X    9  = 

45 

6X    9^ 

54 

7X    9=    63 

8X    9=    72 

5X10  = 

50 

6X10  = 

60 

7X10=    70 

8X10=    80 

5X11- 

55 

6X11  = 

66 

7X11=    77 

8X11—    88 

5X12  = 

60 

6X12  = 

72 

7X12=    84 

8  X  12  =    96 

9X    1  = 

9 

10X1  = 

10 

11  X   1=   11 

12  X    1=    12 

9X    ^--= 

18 

lOX    2:= 

20 

11  X    2=    22 

12  X    2=    24 

9X    3  = 

27 

10  X    3=. 

30 

11  X    3=    33 

12  X    3=    36 

9X    4  = 

36 

lOX    ^  = 

40 

11  X    4=.    44 

12  X    4=   48 

9X    5=. 

45 

lOX      5:= 

50 

11  X    ^^    55 

12  X    5=    60 

9X    6  = 

54 

lOX    6  = 

60 

11  X    ^=    66 

12  X    ^=    72 

9X    7  = 

63 

lOX    7=- 

70 

11  X    7=^    77 

12  X    7=..    84 

9X    8  = 

72 

10  X    8== 

80 

11  X    ^=    88 

12  X    8=    96 

9X    9- 

81 

lOX    9^ 

90 

11  X    9=    99 

12  X      9rrrl08 

9X10  = 

90 

10X10  = 

100 

11  X  10  =110 

12  X  10  ==  120 

9X11  = 

99 

10X11  = 

110 

11  X  11  =  121 

12X11  =132 

9X12  = 

108 

10  X  12  = 

120 

11X12  =  132 

12  X  12  =  144 

MULTIPLICATION.  29 


Write  the  multiplication  table  as  follows: 

2  times  1  or  once  2  is  2. 

2  times  2  are  4. 

2  times  3  or  3  times  2  are  6. 

2  times  4  or  4  times  2  are  8. 
Continue  this  to  12. 

1.  To  find  the  sum  of  123  +  123+123,  we  would 
enter  the  three  amounts  as  in  addition,  and  add  for 
the  result. 

In  multiplication  we  write  123  as  in  the   margin,   and  123 

say,  3  times  3  are  9 ;  put  9  in  the  unit's  place.  3 

Three  times  2  are  0;  put  6  in  the  ten's  place.  

Three  times  1  are  3;  which  put  in  the  hundred's  place.  ggg 
The  result  is  369,  as  it  would  have  been  by  addition. 

TERMS. 

Art.  2.     The  number  123  is  called   the   multipli 
cand,  the  number  3  the  multiplier^  and  369  the  pro- 
duct.    The    multiplicand    and    multiplier    are    also 
called  factors. 

2.  To  find  the  product  of  1496  by  7. 

Here  wo  say  7   times  6  are  42 ;  write  2  under  the  7.  1496 

Then  7  times  9  are  63,  and   the  4  we  carried  make  67 ;  n 

write  7  and  carry  6.     7  times  4  are  28  and  6  are  34;      

write  4  and  carry  3.     7  times  1  are  7  and  3  are  10.  10472 

Ans.  10472. 

3.  2146X2=  4292  4.  21007X  5=* 
3178X3=  9534  31497X  6  = 
4167X4=16668  17843X  7= 
5189X5=*  41679X  8= 
7864X6=  98765 X  9= 
2875X7=  73149X12= 


Total,     123748  Total,     2519023 

*The  pupil  will  fiU  the  blanks. 


80 


MULTIPLICATION. 


Observe  to  point  off  the  cents  in  the  products  of 
the  following: 

6.  $10.78X    9=* 


..  $21.37X  7=* 
117. 49X  8  = 
317. OOX  9= 
671.49X10= 
857.37X11  = 
1096.49X12= 


117. 07X  6  = 
307. 49X  7= 
678.39X11  = 
467.28X12= 
999. 99X   9  = 


Total,  $33246.36  Total,  $25021.08 

7.     2785X357. 
We  have  here  three  multipliers — seven,  fifty,  and 
three  hundred. 
2785X7=  19495  19495 

2785X5  tens=  13925  tens,  or  139250 

2785X3  hundreds=    8355  hundreds,    or    835500 


Total  products, 

This  operation  might  be  contracted  by  arranging 
the  figures  as  in  the  margin,  and  writing  the  first 
figure  of  the  products  of  the  units  in  the  unit's  place 
and  the  others  to  the  left  of  it;  the  first  figure  of  the 
product  of  the  tens  in  the  ten's  place,  or  under  its  own 
multiplier,  5;  and  the  product  of  the  hundreds  in  the 
hundred's  place. 


994245 

2785 
357 


19495 
13925 
8355 

994245 


8.  3170 X   178=       564260 
6184X1794=* 
3867X3784= 
2896X6789= 
7109X9998=  71075782 
2345X3979  = 
6789X2164  = 


1578X   753=^5^ 

9409x6781  = 

2783X4679= 

8976X7659=  68747184 

3968X6483= 

7689X2197= 

6784X7898= 


Total  product,  141049961    Total  product,  242956813 
Note. — Either  factor  may  be  used  as  a  multiplier  in  the  above 


:-The  puj.il   will   fill   the  blanks. 


MULTIPLICATION. 


31 


9.     420001000 
109608 


10.    109608 
420001 


3360008 
2520006 
3780009 
420001 


109608 
219216 
438432 

Product,  46035469608 


Product,  46035469608000 
The  multiplier  of  the  ton's  place  in  the  first  opera 
tion  being  0,  we  passed  it,  and  multiplied  by  the  6 
hundreds.  In  the  second  operation  we  passed  the 
ten's,  hundred's,  and  thousand's  places  for  the  same 
reason. 

NoTE.-^-If  the  learner  will  simply  observe  to  write  the  first 
figure  of  each  product  under  its  own  multiplier,  he  will  have  no 
difficulty  in  multiplying  where  there  are  ciphers.  For  instance, 
the  first  figure  of  the  product  by  2,  in  the  second  example,  is 
immediately  under  the  2.  22. 


11.  12346X  30010=  370503460 
7684X  10900=* 
6787X     3009= 
4967X     6007=    29836769 
5896X900707= 
7649X  66080:= 


2000X  7010=* 

3160X10096= 

2178X90909=  197999802 

1009X90910= 
21678X21006= 
31784X  7009= 


Total,  6320532304  Total,  1013793476 

Art.  3.  To  multiply  by  10,  100,  1000,  etc.,  we  have 
only  to  annex  as  many  ciphers  (  0  )  to  the  multipli- 
cand as  there  are  in  the  multijDlier: 

35X10=350  35 

Explanation. — Multiplying  by  the  1  and  the  0,  we  say  10 

0  time  Sis  0 ;  write  0  under  5,  then  1  time  5  is  5,  1  time     

3  IS  3=350.  350 

Note. —  One  time  the  number  is  simply,  the  number  repeated,  so 
we  may  as  well  annex  the  cipher  to  the  original,  as  above. 


*  The  pupil  will  fill  the  blanks. 


32  MULTIPLICATION. 


165X10     =1650  25X20     =500 

165X100  =16500  25X200  =5000 

165X1000  =  165000  25x2000=50000 

2.  374X10       =  3.  749X2000  = 

268X100     =  836X16000= 

189x1000  =  341X21000= 

267  X  10000=  876  X  92000  = 


Total,  2889540  Total,  102627000 

PRINCIPLES  OF  MULTIPLICATION. 

Art.  4.  When  two  numbers  are  to  be  multiplied 
together,  we  use  for  the  multiplier  that  which  will 
produce  least  figures  in  the  operation.  This  will  be 
accomplished  by  selecting  the  smaller  number,  ex- 
cept where  there  are  many  ciphers,  as  in  Ex.  3. 

Art.  5.  If  a  number  of  articles  and  the  price  of 
one  article  be  multiplied  together,  the  product  will 
be  the  price  of  all  at  the  same  rate. 

If  the  price  of  one  be  in  cents^  the  price  of  all 
will  be  in  cents.  If  in  dollars,  the  price  of  all  will 
be  in  dollars* 

Note. — Cents  are  easily  converted  into  dollars,  by  inserting  the 
separating  point.  Those  on  the  left  will  be  dollars,  the  others 
cents. 

Art.  6.  The  number  of  articles  contained  in  any 
box,  bale,  package,  etc.,  multiplied  with  the  number 
of  boxes,  bales,  etc.,  each  containing  a  like  number, 
will  give  the  whole  number  of  articles  in  all. 

Art.  7.  The  interest,  discount,  premium,  commis- 
sion of  one  dollar  multiplied  with  the  number  of  dol- 
lars, will  give  the  interest  premium,  etc.,  of  the 
whole  number  of  dollars. 


MULTIPLICATION.  33 


Art.  b.  Aii}^  number  multiplied  by  itself,  is  said 
to  be  squared  or  raised  to  the  second  power  —  any 
number  multiplied  by  itself,  and  that  number  again 
multiplied  by  the  first,  is  said  to  be  cubed  or  raised 
to  the  third  power. 

Tllus.— 2X2=4,  or  2d  power  of  2,  or  2^. 

2X2X2=8,  or  3d  power  of  2,  or  2\ 
2X2X2X2=16,  or  4th  power  of  2,  or '2^ 

Art.  9.  Feet  multiplied  by  feet,  yards  multiplied 
by  yaras,  etc.,  produce  square  feet.,  square  yards^  etc. 

Art.  10.  Any  number  of  feet  multiplied  by  the 
number  of  inches  in  one  foot,  will  give  the  number 
of  inches  in  all  the  feet.  Pounds  multiplied  by  the 
number  of  ounces  in  1  pound,  will  give  the  number 
of  ounces  in  all  the  pounds. 

Art.  11.  Halves,  thirds,  fourths,  multi"plied  on 
whole  numbers,  produce  halves,  thirds,  and  fourths. 

1.  What  is  the  price  of  37  bushels  of  corn 
at  37  cents  per  bushel? 

2.  What  should   I  pay  for  357  yards  of 
broadcloth  at  $2.75  per  yard? 

3.  Find  the  cost  of  325  acres  of  land  at  $57 
per  acre? 

Total,  $19522.44 

4.  In  320  bales  of  cotton  there  are  460  lbs.  each, 
how  many  in  all?  Ans,  147200  lbs. 

5.  In  557  pieces  of  muslin  there  are  35  yards 
each,  how  many  in  all?  Ans.  19495yds. 

6.  A  ship  laden  with  flour  has  7950  barrels  on 
board,  and  in  each  barrel  there  are  196  lbs.,  how 
many  pounds  in  all?  Ans.  1558200  lbs. 

*  The  pupil  will  fill  the  blanks. 


34  MULTIPLICATION. 


7.  The  premium  on  a  dollar  is  .03  or  3 
cents,  how  much  on  $149?  * 


8.  At  6  cents  on  the  dollar,  how  much  in- 
terest should  be  received  on  11750? 

Note. — Six  cents  on  the  dollar,  is  the  same  as  6  per 
cent 

9.  At   8    per  cent,   premium,   how   much 
should  be  paid  on  $3764? 


Total,  $410.59 

10.  Find  the  square  of  the  following  numbers : 

37   =  1369  376  * 

570=        324900  219 

10960  109  . 


Total,  120447869  Total,  201218 

11.  103'!  is  how  much? 
107^  is  how  much? 
19'*      is  how  much? 


Total,  1365973 

12.  How  many  square  feet  are  in  a  room 
measuring  15  feet  long,  and  14  feet  wide?     * 

13.  How  many  square  feet  in  a  board  16 
feet  long,  with  an  average  breadth  of  2  ft? 


Total,    242  feet. 

14.  How  many  feet  are  in  573  yards?       Ans.  1719. 

15.  How  many  inches  in  573  yards?      Ans.  20628. 

16.  How  many  rods  are  in  374  acres?  Ans.  59840. 

17.  If  146  is  multiplied  by  f,  what  is  the  product? 

Ans.  438. 

*  The  pupil  will  fill  the  blanks. 

t  The  small   figures  are  called  indices.     They  are  used  to  in- 
dicate to  what  power  the  numbers  are  to  be  raised.    See  Art.  8. 


'tiity 


MULTIPLICATION.  35 


Art.  12. — Practical  Questions:  1.  Find  the  price 
of  87  bushels  of  wheat  at  84  cents  a  bushel. 

2.  If  I  pay  25  cents  a  cwt.  for  freight,  what  should 
I  pay  on  2B07  cwt.  ? 

3.  .What  will  35  acres  of  land  cost  at  $25  an  acre? 

4.  How  many  yds.  of  muslin  in  6  cases,  each  case 
containing  20  pieces,  and  each  piece  35  yds. ;  and 
what  will  be  the  cost  of  the  whole  at  12  cents  a 
yard? 

5.  250  boards  12  feet  long  and  1  foot  broad  were 
sold  at  2  cents  a  foot,  what  was  the  cost? 

6.  A  floor  measures  25  feet  long  and  23  broad, 
how  many  square  feet  does  it  contain? 

7.  A  merchant  failing  in  business  can  pay  only 
37  cents  on  the  dollar,  how  much  will  the  creditor 
receive  to  whom  he  is  indebted  $7587? 

8.  How  many  quarts  are  in  25  bushels? 

9.  In  a  day  there  are  24  hours,  how  many  seconds 
are  there? 

10.  How  many  pints  are  there  in  17  bushels  2 
pecks? 

11.  How  many  inches  are  there  in  3  yards  2  feet? 

12.  In  a  ream  of  paper,  how  many  sheets  are 
there  ? 

13.  A  commission  merchant  receives  2  %,  ov  2 
cents  on  the  dollar,  how  much  should  he  receive  on 
$1425? 

14.  At  $3.75  a  dozen,  what  will  7  dozen  of  chisels 
cost? 

Answers  to  the  above :  $576.75,  $73.08,  $875,  480, 
132,  800,  86400,  1120,  $504,  $68,  575,  $2807.19,  4200, 
$28.50.  $26.25,  $19.50. 


36  DIVISION. 


VI.   DIVISION. 

Art.  1.  Division  is  the  method  of  calcuhition  used 
to  separate  a  number  into  equal  parts. 

Art.  2.  The  sign  is  -h.  When  placed  between 
two  numbers  it  indicates  that  the  one  on  the  left  is  to 
be  divided  by  the  one  on  the  right. 

6-r-3,  reads  six  divided  by  three. 

Art.  3.    Division  is  also  indicated  as  follows: 

3)6.  Which  indicates  that  6  is  to  be  divided  by  3. 

|.  This  is  a  Common  Fraction,  and  indicates  that  6 
is  to  be  divided  by  3  also. 

— — — This  is  also  a  fraction,  and  indicates  that  the 
f  •    fraction  \  is  to  be  divided  by  the  fraction  f . 

.5,  .05  are  called  Decimals^  and  signify  that  the 
first  is  divided  by  10,  and  the  second  by  100. 

Note. — The  separating  point  between  dollars  and  cents  is  a 
decimal  sign,  and  indicates  that  the  figures  on  the  right  are  so 
many  hundredths  of  a  dollar,     $4.25  is  $4y^^^. 

TERMS. 

Art.  4.  The  number  by  which  we  divide  is  called 
the  divisor. 

The  number  to  be  divided  is  the  dividend. 

The  number  produced  by  dividing  is  the  quotient 

The  number  left,  the  remainder. 

dividend, 
divisor  3)167S4 

quotient      5594  —  2  remainder. 


DIVISION.  37 


DIVISION  TABLE. 

The  pupil  can  make  a  division  table  of  the  multiplication 
table,  by  reciting  it  as  follows  :  3  times  2  are  6,  3  in  6,  2  times; 
3  times  3  are  9,  3  in  9,  3  times. 

To  divide  346  by  2. 

Explanation  1. — Commencing  at  the  lefy  we   say  2 

in  3,  1  time  and  1  left;   write  the  1  (time)  under  the  3.  2^346 

2.  Carrying   the   1  that    was   left,  we   suppose   it    to         ^ 

stand  before  the  4,  which  will  raise  that  number  to  14;  X*]^ 
then  2  in  14  7  times ;   write  7  under  the  4. 

4.  Then  2  in  6,  3  times.  Ans.  173. 

Remark. — Until  he  becomes  familiar  with  the  process,  the 
learner  might  write  the  remainders  in  small  figures,  as  in  the  fol- 
lowing example.  When  he  has  mastered  the  operation,  he  can 
dispense  with  them. 

2.  To  divide  13076837617  by  8. 

8)1  3  ^0  27  36  ^8  3  37  ^6  1  ^7 

r  6    3    4    6  0    4    7"^2— 1 

Explanation. — Commencing  at  the  left,  we  say,  8  in  13,  1 
time  and  5  left;  place  5  before  the  next  figure,  then  8  in  50,  6 
times  and  2  left;  place  6  below,  and  2  before  the  7   and  so  on. 

Divide  the  following : 

Quotients.        Rem. 

3.  134615379--2  67307689—1 
21637298452-^3  7212432817—1 
59368217755-4-4                 14842054438—3 

1416823687949--5      283364737589—4 


Rem, 

Rem, 

4.  13645217--6     5 

5.  361745731--10     1 

23176841--7     2 

213764952-f-ll     5 

47896739--8     3 

178961521--12     1 

89765432-^9     8 

345678900--12     0 

Total  quotients,    = 

21546207                =99327785 

Note. — The  value  of  the  remainder  may  be  expressed  in  a  frac- 
tional form:  168-^9  =  18|;  which  signifies  that  9  is  contained 
in  168  eie;hteen  and  six-ninths  times. 


38  DIVISION. 


*6.  Divide  $4537.25  between  7  persons, 

Ans.     Each  person  will  have  $648. 17f. 
Omit  the  remainders  in  the  following  : 

7.  $21372.00^3  8.  $67849132.87^-  8 

13744.00--8  16493178.00--  7 

73176.35--5*  23610934.10-f-  9 

14537.07--9  12310987.47-f-ll 

Total  quotients,  $25092.50,  $14,  579,927.67. 

When  flour  is  $1  a  barrel,  the  loaf  will  weigh  8 
times  as  much  as  when  it  is  $8  a  barrel:  8X9=72 
oz.  at  $1  a  barrel.  At  $6,  it  will  weigh  only  -J  as 
much.     '^g-  =  12  oz. 

9.  If  6  men  do  a  piece  of  work  in  11  days,  how 
long  will  it  take  4  men  to  do  it? 

10.  If  27  men  in  three  days,  do  a  piece  of  work, 
how  long  should  it  take  25  men  ? 

11.  The  interest  on  $367  for  60  days  at  6  %,  is 
S3. 67;  what  should  be. the  interest  on  $1687.25  for 
the  same  length  of  time,  and  at  the  same  rate  per 
cent.? 

12.  In  a  square  foot  there  are  144  square  inches, 
how  many  square  inches  are  there  in  a  room  15  feet  6 
inches  by  18  feet  6  inches? 

Art.  5.  The  quotient  of  a  number  divided  by  2 
is  the  ^  (one-half)  of  it,  divided  by  3  it  is  |,  by  4 
the  {  ;  hence  to  find  the  i,  ^,  |,  etc.,  of  a  number, 
we  have  only  to  divide  by  2,  3,  4,  etc. 

1.  1  of  3716=12381      2.  i  of  34161143764= 1 

Answers:  4880163394f,  4737147654f,  1570^^.  1874. 
7057961,  2880348786,  146|,  1621  3^6^^  306O,  1238|, 
341f,  $648.17-f,  $7124,  $1718,  $14635.27,  $1615.23, 
$16.87,  $8481141.61,  $2356168.28,  $2623437.12,  $1119- 
180.68. 

*  Remove  the  decimal  point  before  dividing,  and  replace  it  in  the  quotient 


DIVISION.  39 


Note. — The  learner  will  observe  that  we  did  not. divide  by  the 
fractions  in  the  preceding  exercises;  on  the  contrary,  we  multi- 
plied by  them.  Omit  the  fractious  in  the  answers  of  questions 
in  the  following  r^'oups: 

3.  9|  times  $14567.85  is  how  much  ? 

Ans.  $135966.60. 
4.  $     345.78X37^  5.  $4563.28X45^,  16^,  18^ 

Total,  $24897.37^.  Total,  $3624130.43. 

6.  If  I  pay  12-|  cents  on  the  dollar  for  a  loan  of 

money,  how  much  should  I  pay  for  the  use  of  $4527  ? 

Note. — On  4527  dollars  I  would  pay  4527  times  as  much  as  on 

1  dollar — 4527  times  12^  cents;  or  12^  times  4527=56587^  cents; 

or  $565.88. 

What  should  T  pay  on  the  following  amounts  at 
the  rates  specified  ? 

7.  $3146@2A,  6A  8.  $71684.25@8i,  7^,  5^ 


1567@3i,  5^  89647.87@U,  2i,  3^ 


7864@ei,  74  79943.57®   {,  6i,  7-1 

Total,  $1474.04.  Total,  $33274.96. 

9.  If  a  steamboat  be  worth  $155367,  what  would 
J  be  worth?!?  1?  I?   ^?  4?  I?    i?  J^? 

Total,  $297108.56. 
Art.  6.    To  divide  by  10,  we  cut  off  the  right  hand 
figure,  then  the  figures  on  the  left  will  be  the  quo- 
tient, and  those  on  the  right  the  remainder. 

1.  Divide  25  by  10.      Operation— 2  \  5,  or  2.5=2f^. 

2.  Divide  6498  by  iOO.    Ans.  64 1 98,  or  64.98^=643%^ 

Note. — We  divide  by  104,  1000,  etc.,  in  the  same  way,  only, 
instead  of  cutting  off  one  figure,  we  cut  off  as  many  figures  as 
there  are  ciphers — for  100,  two  figures;  for  1000,  three  figures,  etc. 

Answers :  $135966.60,  $12966.75,  $1223.50,  $11707, 
125,  $565.88,  364|9.,  2|,  $78.65,  $204.49,  $52.23,  $80.96, 
$491.50,  $566.21,  $5973.687,  $5376.318,  $3942.633, 
$1344.718,  $2241.196,  $3137.675,  $199,878,  $5063.092, 
$5995.767,  $307791,  $206108.14,  $74533.56,  $84420.68, 
$330072.16,  $636567.75,  $582183.44,  $73166.24,  $297- 
108.56. 


40 


DIVISION. 


Note. — If  the  36498  had  been  dollars,  then  the  answer  would 
have  been  $864.98 ;   or  364  dollars,  98  cents. 

3.  43645-^  10  4.  $168938---  10  and  by  100 

71987-^100  678476-f-lOO  and  "      10 

81674^100  396889-f-lOO  and  "1000 

21362-v-lOO  798755-f-lOOand  "      10 

Total,  6114.73  $185444.369 

Art.  7.  When  there  are  cents,  the  division  may 
be  performed  by  removing  the  decimal  jDoint  toward 
the  left.  To  divide  by  10  we  remove  it  one  figure, 
to  divide  by  100  we  remove  it  two  figures,  by  1000 
three  figures,  etc. 

$55.10  ^  10=$5.510,    3167.56  -f-100  =  $l. 67,56. 

Note. — The  value  of  each  and  all  of  the  figures  decreases  ten- 
fold for  every  figure  the  decimal  point  is  removed  to  the  left. 
The  $5  of  first  example  become  50  cents,  and  the  10  cents 
become  10  mills  or  1  cent;  making  the  answer  5  dollars  51  cents  ; 
not  5  dollars  510  cents.  The  second  answer  is  1  dollar  67  cents 
5y6^ths  mills. 

Divide  the  following,  omitting  the  remainders: 


5.  %      457.87- 
1677.45- 


10     6.  $473.04^-1000  and  100 
100         15.17.---  10  and  100 


6109.88---1000         16.57--  100  and   10 
14999.99--  100        106.07-^-  100  and  1000 

Total  answers,  218  dols.  66  cts.  9  dols.  85  cts.  9  mills. 

7.  Divide  the  following  sums  of  money  by  100: 
$645,  $1678.25,  $87493.57,  $16453.27,  $1998.38,  $643.- 
24,  $2168,  $4137.54.  Total  answer,  $1152.16,9 

Art.  8.  It  often  happens  that  there  are  not  as 
many  figures  to  cut  off,  as  there  are  ciphers  in  the 
divisor.  In  such  cases  we  annex  ciphers  to  the  left 
of  the  dividend  to  make  up  the  number. 

Divide  $5.  by  100.  Ans.  .05. 

Explanation. — This  is  the  same  as  removing  the  decimal  point 
two  places  to  the  left,  as  above.     The  $5  had   the  decimal   point 


DIVISION.  41 


on  the  right  of  the  5,  it  is  now  two  places  farther  to  the  left,  and 
therefore  is  divided  by  100.  The  cipher  in  this  case,  as  else- 
where, possesses  no  value. 

8.  $       5-^     10  =  .5  9.  $  .03-^  10 

3-^  100  =  .03  .02-4-100 

4^1000  =  .004  .14-^100 

50-f-lOOO  =  .05  3.16-^100 

457H-1000  =  .457  Ans.  $0.0362 

10.  Divide  the  following  sums  by  100 :  3  cents,  33 
cents,   $3.33,    $33.33,  $333.33,    $3333.33. 

Total,  $37.03,68 

Art.  9.  To  divide  by  20,  300,  5000,  etc.,  we  point 
off  as  many  figures  in  the  dividend  as  there  are 
ciphers  in  the  divisor,  and  divide  by  the  2,  3,  5,  etc. 
The  figures  pointed  off  will  form  part  of  the  re- 
mainder. 

1.  Divide  317745  by  500.  5 1 00)3177  i  45 

635-245 
635  %U 

2.  467831-f-  20=23391i-J  3.  716849-^700= 
716893-^300=2389^^3  897653-^-900  = 
417368-v-500=834|g8  49673-4-  80  = 

Total  quotients,  2083.58 
Note. — To  divide  dollars  and  cents,  first  reduce  the  dollars  to 
cents  or  mills. 

4.  Divide  $36147.59  by  500. 

500)36147159  5 | 00)361475 | 90 

"T2'29^§  ~72295~^o^  miiis 

or,  $72.29  |gg  or,  $72.29,5 

In  the  last  solution  the  answer  is  given  in  dollars, 
cents,  and  mills— 72  dollars,  29  cents  and  5  mills. 
To  reduce  cents  to  mills  we  have  only  to  annex  a 
cipher  to  the  right  of  the  cents,  as  in  this  case. 


42  DIVISION. 


The  answers  to  the  following  are  required  in  dol- 
lars, cents,  and  mills,  omitting  the  remainders: 

5.  $13764.75---50=  -  6.  $16789.37--  80  = 

73968.23-^60=  67859.67-^900= 

37437.18--90=  54168.23^700  = 

18964.20-^80=  78910.00-^-600  = 

7.  Divide  the  following  sums  by  20,  and  give  the 
answers  as  above:  $1367.25,  $3143.57,  $2345.87, 
$34.57,  $45670.44. 

8.  Divide  $34567.25  by  10,  12,  20,  100,  30,  50,  70, 
and  90.  Total,  $11132.84,6. 

9.  Divide  $367897.87  by  100  and  the  quotient  bv 
10,  20,  30,  40,  50,  60,  70,  80,  90.     Total,  $4719.74,4.  " 

10.  Divide  $17654.37  by  100  and  the  quotient  by 
3,  10,  7,  40,  30,  50,  70,  90  and  80.       Total,  $298.78. 

11.  Divide  $314937  by  100  and  multiply  the  quo- 
tient by  7,  then  divide  the  quotient  by  30,  60,  40,  12, 
9,  80,  90.  Total,  $26117.89,9. 

12.  Find  the  sum  of  |-,  i,  J^,  ^\,  ^\,  5^,  J^,  of  the 
yj^th  part  of  $6739.45.  $61.21,3. 

13.  The  ^%  3L,  J^,  ^1^,  ^1^,  ^-i^th  of  $16894  39  divided 
by  100,  is  how  much?  Total,  $41.38,8. 

Art.  10. — Practical  Questions:  1.  At  16  cents  a 
bushel  for  coal,  how  many  bushels  can  be  purchased 
for  127  feet  of  lumber  at  $3.75  a  hundred? 

2.  Three  persons  invest  $6000  in  business.  The 
first,  $3000;  the  second,  $2000;  and  the  third, 
$1000;  and  their  gains  are  $2400;  what  was  each 
man's  share? 

Answers:  $494.16,9,  $2161.11,8,  $11132.84,6,  $2628.- 
53,3,  $4719.74,4,  $26117.89,9,  $298.78,  $41.38,8,  $61.- 
21,3  ;  29f ,  $1200,  $800,  $400. 


EASY    FRACTIONS.  43 


VII.  EASY  FRACTIONS.* 

Art.  1.  A  fraction  is  a  part  or  number  of  parts 
of  any  thing  considered  as  a  whole.  Fractions  are 
of  two  kinds,  common  and  decimal.  A  common  frac- 
tion is  written  with  two  numbers,  called  terms,  having 
a  line  between  them,  as  |-;  a  decimal  fraction  with 
one  number,  having  a  period  at  the  left,  as  .5  (five- 
tenths). 

Art.  2.  A  common  fraction  indicates  division,  the 
upper  number  being  the  dividend  and  the  lower  the 
divisor.  In  treating  of  fractions,  the  dividend  is 
called  the  numerator  and  the  divisor  the  denominator. 

The  denominator  indicates  the  number  of  parts 
into  which  the  whole  is  divided,  and  the  numerator 
the  number  of  such  ^^arts  under  consideration. 

Art.  3.  Value  of  a  Fraction. — The  lowest  value 
of  a  fraction  is  expressed  by  the  figure  1  for  a  nu- 
merator, and  the  highest  value  a  number  as  great  as 
the  denominator  less  l.f  \  represents  the  lowest 
value  of  fractions  of  the  denomination  of  ninths, 
while  -I  represents  the  highest  value  of  that  denomi- 
nation.J 

*This  chapter  is  introduced  for  the  benefit  of  that  large 
class  of  scholars  who  leave  school  before  completing  the  study 
of  Arithmetic.  The  subject  of  fractions  is  treated  of  at  length 
in  the  latter  part  of  this  book. 

TThis  does  not  apply  to  improper  fractions,  which,  as  the 
name  indicates,  are  not  strictly  fractions. 

X 1.  Since  this  is  the  case,  it  is  evident  that  fractions  decrease 
in  value  as  their  denominators  increase,  the  numerators  remain- 
ing the  same.     \  is  less  than  J,  J  than  i,  \  than  J. 

2.  It  is  also  evident  that  the  value  of  a  fraction  depends  on 
the  relation  of  the  numerator  to  the  denominator,  or,  in  other 
words,  the  number  of  times  the  numerator  is  contained  in  the 
denominator,  f  is  equal  to  |,  because  the  numerator  3  is  con- 
tained in  its  denominator,  6,  the  same  number  of  times  that  the 
numerator  4  is  contained  in  the  denominator  8. 


44  EASY   FRACTIONS. 


When  ii  number  is  divided  into  two  parts,  each 
part  is  called  a  half;  into  3  parts,  each 'part  is  called 
a  third;  into  4  parts,  each  part  is  called  a  fourth; 
into  5,  a  fifth;  into  12,  a  twelfth;  into  18,  an  eighteenth; 
into  25,  a  twenty-fifth;  into  100,  a  hundredth;  into 
476,  a  /oi^?'  hundred  and  severity-sixth  part. 

1.  When  a  number  is  divided  into  10  parts,  what 
is  each  part  called?  Into  11?  Into  20?  Into  33? 
Into  45  ?     Into  97  ?     Into  62  ? 

2.  When  divided  into  31,  what?  Into  69?  Into 
103?    Into  364?    Into  155?    Into  1000?    Into  3144? 

3.  Whicli  is  greater,  f  or  f  ?  Ans.  f. 

Reason. — Because  it  will  take  less  to  make  it  a  whole  num- 
ber. The  first  fraction  requires  J  to  make  it  a  whole  number, 
while  this  one  requires  only  J. 

4.  Which  is  greater,  -i  or  -f  ?  |  or  |?  f  or  f  ? 
fori?    |or|?     XorA?    H  ov^?     |or|? 

Since  the  value  of  a  fraction  depends  upon  the  re- 
lation of  the  numerator  to  the  denominator,  [note  2, 
page  43,]  both  terms  may  be  multiplied  or  divided 
by  the  same  number  without  altering  its  value. 

1X2^4  ^^^  2-.2_l 

4X2     8  4-- 2     2 

Now,  I  and  ^  possess  the  same  value  as  f,  because 
their  respective  numerators  are  contained  the  same 
number  of  times  in  their  denominators. 

5.  Change  f  to  twentieths. 

3X5 15        Explanation. — By  multiplying  the  4  by  5,  we 

4\/5      20     change  the  denominator  to  twentieths;  and  by  mul- 
tiplying the  numerator  by  the  same  number  we 
preserve  the  original  value. 

6.  Change  f  to  Sths;  ^  to  12th8;  4-  to  20th8;  |-  to 
14ths ;  f  to  12ths ;  f  to  18ths ;  f  to  30ths. 

A-n«5WPr<5'     75     487        8       1276       6         4        4         0 

15      2  4      3 
18J    30'    4- 


EASY    FRACTIONS.  45 


Art.  4.  To  reduce  a  fraction  to  its  lowest  terms  is 
to  divide  the  numerator  and  denominator  by  such  a 
number  or  numbers  as  will  do  so  without  a  remain- 
der. When  the  terms  can  not  be  exactly  divided  by 
any  number  greater  than  1,  the  fraction  is  in  its  sim- 
plest form. 

1.  Eeduce  ^,  -j^j,  ^,  ^,  -^,  -i^,  to  their  lowest 
terms. 

When  a  single  number  will  not  reduce  the  frac- 
tion, other  numbers  may  be  used,  as  below. 

,    2.  Eeduce  -g^frr  ^^  ^^®  lowest  terms. 

5)9MT(ll)Tik(TiT- 

3.  Eeduce  to  their  lowest  terms,  -^^j  -^^^  ^|^,  and 

Teir- 

Fractions  may  be  Proper^  Improper^  Simple^  Com- 
pound or  Complex.  We  shall  treat  of  only  the  three 
former  at  present. 

A  proper  fraction  is  one  whose  numerator  is  less 
than  its  denominator,  as  ^,  An  improper  fraction  is 
one  whose  numerator  is  equal  to  or  greater  than  its 
denominator,  as  f,  ^. 

Art.  5.  A  simple  fraction  is  a  single  fraction,  and 
may  be  proper  or  improper,  as  f,  f . 

Art.  6.  When  a  whole  number  and  fraction  ap- 
pear together,  they  are  called  a  mixed  number,  as  5|. 

Art.  7.  Improper  fractions  may  he  changed  to  whole 
or  mixed  numbers  by  dividing  the  numerator  by  the  de- 
nominator.'^ 

^Answers:  3^,  ^,  ff,  ^-^,  ^%  2f,  ^,  i    -L,'^, 

TT)    Tj  TT- 

*This  is  simply  acting  on  the  principle  that  the  numerator 
is  the  dividend  and  the  denominator  the  divisor. 


46  EASY    FRACTIONS. 

To  change  ^  to  a  mixed  number. 

5)13        Explanation. — There  are  5  fifths  in  one  whole  num- 

23  ber;  in  13  fifths  there  are  as  many  Is  as  the  number  of 

times  5  is  contained  in  13,  which  is  two  times,  with 

3  fifths  over,  making  2|. 

1.  Change  |,  f ,  f,  -^,  ^-^-,  %4_  to  whole  or  mixed 

numbers. 

Art.  8.  To  change  whole  or  mixed  numbers  to  im- 
proper fractions  is  an  operation  the  reverse  of  the  last, 
which  scarcely  needs  explanation. 

1.  Change  9|-  to  an  improper  fraction. 

9f  Explanation. — In  1  wliole  number  there  are  5  fifths; 
_5  in  9  there  are  9  times  5  or  5  times  9  fifths,  to  which  we 
i9     add  4  fifths,  and  we  have  -Y-. 

2.  ^Change  the  following  mixed  numbers  to  im- 
proper fractions  :  3|,  9^  8f,  5f ,  41f ,  97i  16|. 

Art.  9.  To  multiply  a  fraction  by  a  whole  number 
is  simply  to  multiply  the  numerator  without  alter- 
ing the  denominator,  or  to  divide  the  denominator 
without  altering  the  numerator. 

To  multiply  -^  by  6. 

12      -11-^^  ^"^  ^2- 

Eeason. — Assuming  that  7  is  a  whole  number,  multiplying 
it  by  6  gives  42 ;  but  since  it  is  not  a  whole  number,  but  twelfths^ 
the  42  is  f|=3r«2  or  Z\. 

By  decreasing  the  denominator,  the  fraction  is  in- 
creased (as  it  takes  fewer  of  the  small  parts  to  make^ 
a  whole  number)  ;  hence,  the  7  represents  halves  in- 
stead of  twelfths.     |-=3-i-. 

Answers:  H,  1^,  If,  3,  30^,  69^  ^,  ^,  -^S  ¥.  ¥. 
^,  ^,  ¥.  3i,  h  ¥,  69,^. 

*  The  learner  should  prove  the  accuracy  of  his  work  by  last 
article. 


EASY    FRACTIONS.  47 


1.  f  X7-?        3.  ^9^x12  =  ?        5.  i|xll  =  ? 

2.  IX9-?        4.  ifx6^?        6.  1^X12  =  ? 

Art.  10.  To  multiply  a  whole  number  by  a  fraction^ 
we  multiply  the  numerator  without  altering  the  de- 
nominator. 

1.  Multiply  25  by  |. 

25X3  fourths=75  fourths,  or  -"^,  which,  changed  to  a  mixed 
number,  [Art.  7]  =18f. 

2.  35xf  =  ?     3.  134X2^--?     4.  16xA  =  ? 
Art.  11.    To  multiply  a  mixed  number  by  a  ichole 

number. 

Multiply  7f  by  9. 

7S        Explanation. — 3  fourths  multiplied  by  9—27  fourths, 
9      or  6f ;  and  the  7  multiplied  by  nine=6o,  plus  the  6=69, 
QQ3     making  the  product  69|. 

Or  thus :  7| 

A 
3^1X9  =  ^^  =  691 

1.181x5=?     2.  29fx8  =  ?     3.83^x7  =  ? 

Art.  12.  To  multiply  a  whole  member  by  a  mixed 
number. 

1.  Multiply  29  by  8|. 
29 

8f        Explanation.— Multiplying  29  by  2  thirds,  we  have 
,qi     58  thirds,  or  19J,  which  we  write  in  the  first  line.     Then 
232^    29X8^232,  which,  added  to  19J=251J. 

25ii 
Or  thus :  29X¥=^f^=251  J. 

2.  15x3|=:?      3.  12X12^  =  ?      4.  14xl7f=? 

Answers:  ^,>f,  lOf,  4if,  9^,  10||,  28,  6^,  4f, 
69f,  60,  150,  6f,  93|,  238,  5831,  20,  251i  150,  246|, 
18f,  53|,  15|,  72^,  5f. 

The  Teacher  will  find  it  important  to  require  the  learner  to 
preserve  the  process,  as  he  will  be  apt  to  adopt  clumsy  methods 
of  solution. 


48  EASY    FRACTIONS. 

To  multiply  a  fraction  by  a  fraction. 

3.  Multiply!  by  f 

Assuming  the  numerator  5  to  be  a  whole  number,  -|X5=-^^; 
but  5  is  not  a  whole  number,  but  5  sixths;  lience  -^^  is  6  times 
too  much.     •  j^-  divided  by  6=i|,  or  f .     [Note  1,  page  43.] 

Art.  13.  Hence,  to  multiply  a  fraction  by  a  frac- 
tion, we  multiply  the  numerators  together  for  a  new  nu- 
merator,  and  the  denominators  for  a  new  denominator. 

|X|-=Yfj  which,  reduced  to  its  lowest  terms  =|-.* 

2.  fXf-?     .      3.  ^X|-?  4.  iXli-? 

Art.  14.  To  midtiply  a  mixed  number  by  a  fraction 
or  a  mixed  number. 

1.  Multiply  15|  by  -^. 

l^=:-^f,  which,  multiplied  by  -jSj^^  W  ^i^  l^^- 

2.  Multiply  8f  by  16f . 

83^3^  and  16|=^.     ^x^=^fF=145f|==145f 

3.  12iXl6t---?     4.  14fx-i^-?     5.  18i-Xl2i  =  ? 

Art.  15.  To  divide  a  whole  number  by  a  fraction  or 
a  mixed  number. 

1.  Divide  315  by  f,  or,  in  other  words,  find  how 
often  I  is  contained  in  315. 

Solution. — Before  we  can  measure  315  by  fourths,  we  must 
change  it  to  fourths.     In  1  there  are  4  fourths ;  in  315  there 

Answers:  fi,  f,  |i,  14^,  146|,  208^,  12f  231^,  |, 

2^^ 

*It  will  be  observed  tliat  to  multiply  by  a  fraction  does  not 
increase  the  multiplicand,  as  in  whole  numbers;  but,  on  the 
contrary,  decreases  it,  the  f  being  less  than  J. 

To  account  for  this,  it  is  only  necessary  to  remember  that  a 
whole  number  is  reduced  to  the  denomination  of  a  fraction  by 
being  multiplied  by  it.  6X1^18  fifths,  or  3f.  Much  more  is 
a  fraction  reduced  in  value  if  multiplied  by  a  fraction.  From 
this  we  readily  infer, 

2.  That  to  divide  by  a  fraction  increases  the  dividend. 


EASY    FRACTIONS.  49 

are  315  times  4  or  1260  fourths,  which,  divided   by  3=430. 
Hence,  f  is  contained  in  315  420  times. 

Operation.    315    or  ^^Xi-=-T--=^^0 
4 


3  j  1260 
420       . 

2.  320-v-|-=?      4.  541-f-|=? 

3.  27-- 1  =  ?      5.  684--^=? 

6.  987- 

7.  136- 

8.  Divide  25  by  Sf 

Operation.    25X2   l)alves=-\o   and 

5|X2=Y. 

50^11 

4A,  or  ¥XA-=W=^4^|=43^. 

Art.  16.  Hence,  to  divide  by  a  fraction,  we  multi- 
ply by  the  denominator  and  divide  by  the  numera- 
tor, or  invert  the  divisor  and  proceed  as  in  multipli- 
cation. 

1.  157-f-  3i-=?      4.  345--  6f=-?.     7.  195-f-16|=? 

2.  22-f-12|=?       5.     39---15i=?       8.     39--124rr=? 

3.  16--16f=.?       6.     79-f-37^=?       9.     87--3l|^  ? 

To  divide  one  fraction  by  another. 

10.  Divide  I  by  f.     Operation.  fx|-=i|=i^. 

Explanation. — By  inverting  the  divisor,  we  obtain  |§,  the 
terms  of  which,  being  divided  by  2,  give  ^. 

11.  t2j^|=?      14.  31i--f  =?      17.   f -^li=? 

12.  T%-^i-  ?      15.  31f-^=  ?      18.  T%^3i=  ? 
13-  li=f=?      16.  13|-|=?      19.    5^6f=? 

Art.  17.  To  divide  when  either  divisor  or  dividend  is 
a  mixed  number  and  the  other  term  a  whole  number^ 
both  terms  may  be  reduced  to  the  same  denomina- 
tion.    [Art.  15.] 

I.  Divide  34571  by  13. 

Answers :  32|,  265f ,  2434|,  19740,  2280, 151|,  24f , 

1  1    iLO.    JL9     Q«2  9     Qf?3     IfiXJLO       9       7     1      IO4.J.6     119     64 

511     2-25     9  8       517     ^3       2      2     998 
•^■^ir>  *^¥6>  ^1^)  ^TO"?  ^6  4>  Zi  T?  ^T25- 


50  EASY    FRACTIONS. 


3457J  Explanation. — The  dividend  containing  the 

4  fraction  of  J,  both  terms  are  reduced  to  fourths, 

62)13829(565     ^^^  division  performed  as   in  whole  numbers. 

104  '-The  result  shows  that  the  divisor  is  contained 

•  oAo  ^^  ^^1^  dividend  265  times,  with  a  remainder  of 

Q19  49  fourths  [Art.  2,  Principles  of  Division,  page 

^^  46],  or  26511  times. 

269 
49 

The  same  by  short  division, 

13)3457  J  Explanation.— 13  is  contained  in  3457  265 
"25549  times,  with  a  remainder  of  12,  which,  reduced 
^^  to  fourths,  including  the  J  of  the  dividend,  is 
49  fourths.  13  not  being  contained  in  this  an  even  number  of 
times,  the  denominator  is  increased  13  times,  (which  is  the 
same  as  to  decrease  the  numerator,)  which  gives  the  same 
fraction  as  by  long  division,  |f . 


2.  1398|  :  56  — -  ? 

5.  1255|  :  350—? 

3.  2564  :  7  — ? 

6.  7961:421—? 

4.  1939  --  8^=? 

7.  467|--  12=? 

Art.  18.  To  subtract  a  fraction  from  another  of  the 
same  denomination  is  simply  to  subtract  the  less  nu- 
merator from  the  greater. 

1.  From  ^  take  -^. 

Q      SI 2_5 — 9 

^'    ¥2        42 —  ' 

4     i_6 5_ ? 

^•22        22 • 

Art  19.  To  subtract  a  fraction  or  mixed  numoer 
from  a  whole  number. 

1.  From  9  take  3f. 

Answers:  275|f,  24^,  36^^^,  3|4^,  1^1  232i|, 
38f^,  h  h  h  iV>  h  h  h  h  A.  A^  tV. 


EASY    FRACTIONS.  '  51 

The  following  formula  will  render  the  operation 
simple : 

Whole  number.  Fourths         EXPLANATION. — Arranging  the  less  un- 

,  ^*  der  the  greater,  we  find  we  can  not  take  3 

fourths  from  0  fourths;  so  a  whole  number 

.  "T  "T  or  1  is  added  to  botli  terms.     In  1  there 

^  ^j  are  4  fourths,  from  which  we  take  3  fourths, 

^'*    ?  and  we  have  a  remainder  of  1  fourth.     To 

the  3  add  1  and  we  have  4,  which,  subtracted  from  9,  leaves  5, 

giving  for  the  answer  5|. 

2.  13—  41=.=  ?      5.  11-2  i=?      8.  52— 27i^? 

3.  15—  b\=l      6.     7—     |=r?      9.  13—121:^? 

4.  29-121=  ?      7.  14—1^=  ?    10.  89—75^=  ? 

Art.  20.  To  subtract  one  fraction  from  another  of  a 
different  denomination,  it  will  be  necessary  first  to 
reduce  both  to  the  same  or  a  common  denominator. 

1.  From  I  take  f 

By  Art.  3,  |-  can  be  changed  to  56ths  by  multi- 
plying both  terms  by  7,  and  ^  can  be  changed  to 
56ths  by  multiplying  both  terms  by  8,  giving  -||  and 
1^,  the  difference  between  which  is  -^-^^  the  answer. 

It  will  be  observed  that  the  multipliers  used  in 
this  case  were  the  denominators,  7  and  8,  which, 
multiplied  together,  give  a  common  denominator^  and 
multiplied  into  the  numerators  of  each  other  give 
the  new  numerators. 

Operation:    |-^=ff-4|_^. 

2.  From  f  take  \  .         5.     |— 1=  ? 

3.  From  |  take  |.    '     6.  6^ — fz=z?         9. 

4.  From  |  take  ^. 

Art.  21.  To  add  fractions  of  the  same  denomination, 
the  numerators  only  are  added,  and  the  sum  reduced 
to  a  mixed  number  or  its  lowest  terms 

Answers:  |,  61   9f,  41,  24f  8|,  8|,  16|,  13f,  13f, 

5       53.     23      5.    _3_2       1      23      1       5       3     193       1 
T¥:  *^4'    3Tj    8'    193^'    6"'  T2"'  ¥'  "2  8"?  ~E'  ^^t'  T2* 


52  EASY    FRACTIONS. 


1.  Add  l+f+l+f 

3  Explanation.— Here  the  four  numbers  are  added 

6         together,  making  21  eighths,  which,  reduced  to  a  mixed 
5         number,  are  equal  to  2f. 


2-  iV+A+A+A+A+i^+A=? 

Art.  22.  To  add  fractions  of  different  denominations ^ 
they  should  first  be  reduced  to  a  common  denomi- 
nator, as  in  subtraction. 

X_I3.: — ?  i.4_3. 4.16 10 12^^11 

When  three  or  more  fractions  of  diiferent  denomi- 
nations are  to  be  added  together,  they  may  be  re- 
duced to  a  common  denominator  by  multiplying  all 
the  denominators  together,  as  above,  and  then  by 
multiplying  each  numerator  by  all  the  denomina- 
tors except  its  own.* 

1.  Find  the  sum  of  ^-+1 +|. 

2  x4x  6^=^48r:r:(7ommon  denominator, 
1 X 4X 6=24=:i^zr5i^  numerator. 
3x2x6=36=AS'econd  numerator 
5  X  2  X  4=40=  Third  numerator. 
100=Sum  of  numerators. 
Hence,  0^=2^=2,^. 

The  J  in  the  example  was  multiplied  by  24,  giving  |f ;  the 
f  by  12,  giving  ff ;  and  the  |  by  3,  giving  ff. 

2.  f +|=?       5.    |+t+|-=?       8.  2f+  I+I-? 

3.  i+i=?       6.    ^+i+i  =  ?       9.  5i+6i+|  =  ? 

4.  A+I-?       '^.  A+i+A=?     10.     |+2|+^=? 


Answers :  2fT  3^,  UfJ^  ^J^,  H,  4X,  12,  3^^,  H, 

05    12?    2'   ^24- 


*This  is  simply  multiplying  both  terms  by  the  same  number. 


DECIMALS.  53 


VIII.  DECIMALS. 

Art.  1.  A  Decimal  Fraction  expresses  its  value 
in  one  term,  and  is  known  from  a  whole  number  by 
its  having  a  period  called  a  decimal  point  at  the  left. 
.5  is  a  decimal. 

The  value  of  a  decimal  is  more  easily  ascertaine( 
than  that  of  a  common  fraction  ;  while  operations  in 
decimals  are  performed  with  nearly  the  same  ease 
as  those  in  whole  numbers. 

Art.  2.  The  numerator  only,  of  a  decimal  frac- 
tion is  written,  the  denominator  always  being  10  or 
some  power  of  10.  A  decimal  composed  of  one 
figure  as  .5,  will  have  10  for  a  denominator.  One 
composed  of  two  figures,  as  .75,  will  have  100  for  a 
denominator,  etc.;  hence  to  find  the  denominator, 
we  have  only  to  write  as  many  ciphers  as  decimals, 
and  annex  a  1  to  the  left. 

Remark. — Ciphers  on  the  extreme  right  of  a  decimal  possess 
no  value.     .500  is  the  same  as  .5. 

The  value  of  .073  is  how  much,  expressed  as  a 
common  fraction  ?  Arts.  jj|^ 

What  is  the  fractional  value  of  the  following: 

.23,  .007,  .013,  .75,  .12,  .11,  .4,  .42,  .710,  .0076, 
10.7? 

Art.  3.  The  remoyal  of  the  decimal  point  one 
place  to  the  right  or  left,  increases  or  diminishes  the 
fraction  10  times. 

31.67X10=316.7,  and  316.7-^-10=31.67. 

ADDITION  AND  SUBTRACTION. 

Art.    4.     Operations  in  Addition  and  Subtraction 

of  decimals  are  performed  in  the  same  way  as  those 

in  addition  and  subtraction  of  simple  numbers.    The 

pupil  who  has  calculated  Federal  money,  is  already 


54  DECIMALS. 


acquainted  with  these  operations.  He  has  only  to 
observe  that  units  be  placed  under  units,  etc.,  or 
what  is  still  more  simple,  to  place  the  decimal  points 
directly  under  each  other,  and  proceed  as  in  simple 
numbers.  1.07  +.001  +  37.045  +  10.06+.0007  would 
be  done  thus: 

1.07 
.001 
37.045 
10.06 
.0007 

48.1767 

1.  Find  the  sum  of 

.007+31.06+  .1009+3  00.07=* 
710.34+2.406+67.709+  .0006  = 
314.60+.0006+  .0027+     .001  = 
714.06+  .003+  8.007+     800.= 


2.  Total,  2748.3678 
123.006+.18532+.0185+1672.3+1865.01=* 
184.003+.0185+11.10+18639.01+1657.003= 
0.005+2683.17+2.95+6892.02+.0031= 

Total,     33729.80242 

3.  Find  the  difference  between  107.06  and  .213. 
617.07—41.7106=*  107.06 

10.06—     .9092=  .213 

illSt  ?S:_ 

Total,  725.7592 

4.  21.80— .0503=* 
364.2—128  9= 
6.295—2.654=    

Total,    260.6907 
Note. — For  Muhiplication,  Division,  and  Reduction  of  De- 
cinaals,  see  chapter  XXXIII,  pages  173-176. 

':--  The  pupil  will  fill  the  blanks. 


'SHORT    METHODS.  55 


IX.  SHORT  METHODS. 

PROPERTIES  OF  NUMBERS. 

Art.  1.  Numbers  ending  with  5  or  0  are  divisible 
by  5  without  a  remainder. 

Art.  2.  If  the  two  right  hand  figures  of  a  number 
are  divisible  by  4  without  a  remainder  the  whole 
number  is  divisible  by  4. 

Art.  3.  If  the  three  right  hand  figures  of  a  num- 
ber are  divisible  by  8  without  a  remainder  the  whole 
number  is  divisible  by  8. 

Art.  4.  If  the  sum  of  the  figures  of  any  number 
be  divisible  by  3  or  9  the'whole  number  will  be  di^ 
visible  by-B  or  9. 

SHORT  METHODS  OF  MULTIPLYING. 

Art.  5.  An  even  part  of  10,  100,  1000,  etc.,  can 
be  multiplied  mentally  by  division. 

ALIQUOT  PARTS    OF   100.  ALIQUOT    PARTS  OF   1000. 

50  =i         14|=4  333i=  1 

331=1         121=^  250=^ 

25=1         10  =-1,  166i=^ 

20=4  81=  J.  125=1 

^n  =  h  H  =  i\  83i  =  J^ 

To  multiply  by   a  part  of  100,  we   suppose  two 
ciphers  to  stand  at  the  right  of  the  number  and  di- 
vide by  the  part.     To  multiply  by  a  part  of  1000,  we 
suppose  three  figures,  etc. 
1.  To  multiply  176  by  12^. 

Operation  8)17600 
Arts.      2200 


56  SHORT    METHODS    OF    MULTIPLYING. 

2.  To  multiply  379  by  250. 

Operation  379000^^^^^^  ^^^^ 
4 

3.  To  multiply  $49.75  by  125. 

Operation  4975000     ^^,  ^^, 
^  — ^ =621875  cents, 

®    or  S6218.75  Ans. 

Note. — Only  a  few  answers  will  be  given  in  the  following,  as 
the  pupil  can  prove  the  accuracy  of  his  calculations  by  multi- 
plying in  the  ordinary  way. 

4.  $140  X    12^=$1750  7.  949   X333J= 

5.  3767X      81=  8.  179  X      2i= 

6.  9987  X   25  =  9.  769   X     3|= 

10.  675  yards  @  37-J-  cents. 

Operation   675  at  a  dollar =$675. 00 

at  25c.  =\  168775 
at  121c.  =1       84.371 

Ans.  2537121^ 

11.  715X621  cents.  14.  9876 X $2. 18| cents. 

12.  947X871  15.     719X   3.621 

13.  194Xl8f  16.     965X   4.37^ 

I^OTE. — The  multiplier  of  the  12th  Ex.  wants  only  12J  cents, 
or  J  of  being  a  dollar;  so  we  find  the  cost  of  947  at  a  dollar, 
and  take  off  ^  of  it.  Other  examples  may  be  solved  in  the 
same  way. 

Art.  6.  To  find  the  cost  when  there  are  fractions 
in  both  factors  :   18f  lbs.  @  12i  cents. 

Operation,     18f  lbs.  @  a  dollar  :=  $18.75 

at  121  cents  =1  or    $2.34  Ans. 
2.  37|-  lbs.  @  18f  cents. 

Operation,     37i  @  a  dollar  irr:  $37.50 

at  121:==  lor      4.687 
"     6^  =  1   ^'       2.343 

Ans.    $7.03 


SHORT    METHODS    OF    MULTIPLYING.  57 

1.  To  multiply  424  by  97  : 

Operation.     424X100  =  42400 
424X     3=  1272 


41128 
Art.  7.  When  the  multiplier  wants  from  1  to  12 
of  being  100,  200,  3000,  etc.,  the  work  can  be  con- 
tracted by  multiplying  by  one  of  these,  and  subtract- 
ing as  many  times  the  multiplicand  as  the  multiplier 
is  short  of  them. 

Art.  8.  When  the  multiplier  is  29,  39,  49,  etc., 
we  multiply  by  the  next  higher  number  and  subtract 
the  multiplicand. 

1.  To  multiply  176  by  59  : 

Operation.    176X60  =  10560         2.  671 X   39 

176  3.     59X689 

TT—-         4      89X784 

^""^^^         5.  167  X   29 

Art.  9.     To  square  numbers  that  end  with  9. 
What  is  the  square  of  29? 
Operation.   29 
29 

841 

ExPLAx\AT[oN. — Writing  1  for  the  first  figure  of  the  product,  we 
add  1  to  the  ten's  place  of  the  multiplier,  and  multiply  the  sum 
on  the  multiplicand  less  1:  3X'^8^84,  with  the  1  annexed  =841. 

Find  the  square  of  the  following  numbers  mentally : 

99,     59,     119,     79,     19,     69,     129,     89. 

Art.  10.  To  square  any  number  of  9*s  instantane- 
ously, and  without  multiplying. 

Commencing  at  the  left,  we  write  as  many  9's  less 
one,  as  the  number  to  be  squared,  an  8,  as  many  O's 
as  9's,  and  a  1. 

The  square  of  9999999  is  99999980000001. 


58  SHORT   METHODS    OF   MULTIPLYING. 

Remark. — The  square  of  any  number  of  3*s  will  be  the  ^  of 
the  square  of  the  9's. 

Art.  11.  To  7nultiply  by  375,  625,  750,  or  875,  we 
first  multiply  by  125,  (Art.  5,  Ex.  3,)  and  that  pro- 
duct by  3,  5,  6,  or  7. 

1649X625  =  ^-^l^^=??i^=1030625 

o  o 

Art.  12.    To  square  numbers  under  135,  ending  with  5 

The  first  two  figures  on  the  right  of  the  product 
will  always  be  25  ;  and  to  find  the  others,  we  add  1 
to  the  ten's  place  and  multiply  it  on  the  ten's  and 
hundred's  places  above. 

To  square  115  :  Operation.     11 

12 


13225 

Art.  13.    To  square  a  number  containing  a  half,  aH 

12^,  we  multiply  the  whole  number  by  the  next  higher 

number  and  add  a  fourth.    8J  squared =8X9+^=72^. 

Art.  14.    To  multiply  hy  numbers  of  two  figures 

containing  the  figure  1. 

1.  Multiply  346  by  15.  Operation.  346 

1730 

5190 

2.  Multiply  346  by  51.  Operation.      346 

1730 


17646 


These  operations  might  be  performed  mentally. 
Taking  the  2nd,  for  instance,  we  say  one  time  6=6 ; 
then  5  times  6=30  and  4  of  the  upper  line  =34;  5 
times  4=20  and  3  of  the  upper  line  and  3=26  j  5 
times  3=15  and  2  =  17.     Product  17646. 


SHORT   METHODS   OF   DIVmiNG.  59 


SHORT  METHODS  OF  DIVIDING. 

Art.  15.  To  divide  by  a  number  composed  of  two  or 
more  factors j"^  as  96.  which  is  comiDOsed  of  the  factors 
12  and  8,  or  G48  which  is  composed  of  9X8X9.  This 
operation  is  performed  by  using  the  factors  instead 
of  the  whole  number. 

1.  Divide  $7854  by  32.  Operation.  4)7854 

8)1963—2 
How  the  true  remainder  is  found :  245 — 3 

The  first  remainder  is  2  dollars,  because  it  is  left 
from  the  dollars  that  were  divided.  The  second  re- 
mainder is  four  times  as  great  as  if  it  were  from  the 
first  line,  because  every  figure  of  the  second  line  is 
four  times  as  great  as  if  it  stood  in  the  first  line. 
Four  times  3  =  12,  and  the  2  of  the  first  remainder 
equal  14,  the  true  remainder.  Ans.  245||. 

2.  Divide  6371  by  336.  Operation.  6)6371 

Remark. — The  true  remainder  of  this  example  ^ 

is  found  by  multiplying  the  last  remainder  by  7,  g^    \p^\ 4 

to  make  it  of  the  same  value  as  if  it  were  from  

the  line  above,  and  that  by  6,  to  make  it  of  the  18-^7 
same  value    as  if  it   were    from  the  upper  line: 
7X7X6=294,    to    which,    add    6X4+5    or   29. 
The  true  remainder  is  323.                   Ans.  18|||. 

3.  Divide  1463  by  28      7.  4571--441 

4.  "   7714  '*  72      8.  1987--378 

5.  '*   1943  ''     49       9.  9843--720 

6.  "       8765  **   343  10.  1456-f-729 

Answers :  103f ,  25^1 ,  *5^,  IfH.  52i,  39f|,  lOif, 

*  Such  numbers  are  called  eomposUe.    When  a  number  can  not  be  di- 
vided into  factors,  it  is  called  a  prime  number. 


60  SHORT   METHODS   OF   DIVmiNG. 


DIVISION  BY  CANCELLATION. 

Art.  16.  To  cancel,  signifies  to  blot  out  or  make 
void.  Division  by  cancellation  is  performed  by  writing 
the  terms  in  fractional  form,  and  dividing  them  by  any 
number  that  will  do  so  without  a  remainder. 

1.  To  divide  1463  by  28. 
209 

4 

Explanation. — The  first  terms  were  canceled  by  dividing  by  7, 
leaving  ^J^,  which  finished,  is  52J. 

2.  3465-!-  35  5.  1962-^  22 

3.  2763--  81  6.  6876^-152 

4.  6545-^245  7.  5436-^-144 

Art.  17.    To  divide  by  aliquot  parts  of  100,  1000,  etc. 

This  process  is  the  reverse  of  that  under  Art.  62, 
page  69. 

1.  To  divide  7654  by  25.     Operation.  76.54 


306,16=306^% 

Note  1. — The  decimal  value  of  the  remainder  is  always  obtained 
first  by  this  process.  If  we  divide  the  16  by  the  multiplier,  it  will 
give  the  true  remainder  or  4. 

2.  If  the  aliquot  part  is  of  100.  there  will  be  two  places;  if 
1000,  there  will  be  three  places  of  decimals. 

2.  To  divide  19765  by  125.  Operation      19.765 

8 


158.120 

158. ^\ 

3.  17630^33i,  74910-r-12^,  87396^-125,  824--2i. 


or  158. y\A^ 


PERCENTAGE.  61 


X.  SIMPLE  PERCENTAGE. 

Art.  1.  Percentage  embraces  all  those  calcula- 
tions in  which  100  is  made  the  basis  of  comparison. 

Art.  2.  Fer  cent,  signifies  by  the  hundred.  6  per 
cent,  signifies  6  for  every  100.  ^  is  the  sign,  and 
is  written  thus  :  6  ^ ,  which  reads,  six  per  cent.  The 
6  is  called  the  rate. 

Art.  3.  One  per  cent,  of  a  number  is  that  number 
divided  by  100.  One  per  cent,  of  $320  is  $3.20.  To 
find  any  other  percentage  of  a  number,  we  multiply 
one  per  cent  of  it  by  the  rate. 

Art.  4.  Percentage  may  be  divided  into  simple 
and  complex. 

Art.  5.  Simple  Percentage  embraces  all  those 
calculations  in  which  both  the  principal  and  the  rate 
are  known  and  is  applied  to  Premium,  Discount, 
Exchange,  Taxes,  Commission,  Brokerage,  Insurance, 
Insolvency,  Loss  and  Grain,  etc. 

1.  Find  6%  of  758.  7.58=1  per  cent. 

6 

45.48=6  per  cent. 

2.  3%  of  215=6.45.  3.  9%  of  $756= 
4.  4%  of  788=                   5.  Ifo  of  $179= 

Note  1. — To  find  1^  of  the  following,  remove  the  dec.  point 
two  places  to  the  left,  and  point  ofi^  four  figures  in  the  product. 

2.  Give  your  answers  in  dollars  and  cents.  If  you  have  6 
mills  or  more,  add  a  cent  to  the  cents;  if  less  than  5,  omit  them, 

6.  $768.15@6%=  7.  $566.75@13  %  = 

8.  $196.55@5%=  9.  $789.15@33J%  = 

Note. — When  the  percentage  of  sums  under  $10  is  required, 
it  will  be  more  simple  first  to  multiply  by  the  rate  and  then 
divide  by  100. 

10.  25^  of  $0.97=  11.  50  %  of  $0.17= 

12.  65%  of    0.08=  13.  33J%  of    0.76= 


62  PERCENTAGE. 


Art.  6.  Premium  is  a  percentage  to  be  added; 
discount,  a  percentage  to  be  subtracted  from  the  face 
or  par  value  of  a  bill,  note,  etc. 

Art.  7.  The  method  of  settling  accounts  between 
persons  in  distant  places  by  draft  is  called  Exchange, 
If  between  persons  in  the  same  country  it  is  called  Iti- 
land  or  Domestic  Excliange. 

Art.  8.  Exchange  may  be  par,  or  it  may  be  at  a 
'premium  or  at  a  discount. 

1.  What  will  be  the  cost  of  a  draft  ©n  New  Orleans 
for  $3200  at  1%  discount? 

2.  What  amount  will  pay  for  a  bill  of  exchange  on 
New  York  for  $1860  at  f  %  premium? 

3.  Ax  ^%  discount,  what  should  I  receive  for  New 
York  exchange  calling  for  $728  ? 

4.  At  ^%  premium,  what  will  a  bank  pay  for  a  draft 
on  Chicago  for  $276  ? 

Art.  9.  Commission  or  brokerage  is  the  percent- 
age charged  by  a  commission  merchant,  factor,  agent, 
or  broker,  for  transactii^g  business  for  another. 

Art.  10,  Commission  is  usually  reckoned  on  the 
whole  amount  of  sale,  purchase,  or  collection. 

1.  At  2^  per  cent.,  what  is  the  commission  on 
$17640? 

2.  A  merchant  sells  goods  for  another  to  the 
amount  of  $4371.87,  what  is  his  commission  at  5  per 
cent  ? 

3.  A  broker  receives  \  per  cent,  for  selling  $2500 
worth  of  merchandise  for  a  commission  merchant, 
what  was  the  amount  of  his  brokerage? 

4.  A  of  New  Orleans  buys  sugar  for  B  of  Cincin- 
nati, to  the  amount  of  $7100,  what  is  the  amount  of 
his  commission  at  1^  per  cent.  ? 

Answers:  $441.00,  $218.59,  $6.25,  $106.50,  $3192.00, 
$1866.98,  $727.09,  276.69. 


PERCENTAGE.  63 


Art.  11.  Insurance  is  a  guarantee  against  the  loss 
of  property  by  fire  or  the  dangers  of  transporta- 
tion. 

The  amount  paid  for  guarantee  is  called  the  Pre- 
mium. It  is  a  certain  percentage  on  the  estimated 
value  of  the  property  insured.  This  percentage  or 
rate  is  large  or  small,  in  proportion  to  the  risk. 

1.  How  much  should  be  paid  to  insure  a  bouse 
valued  at  $1674,  premium  li%,  and  policy  $1.50? 

2.  At  2\  %  premium,  wh^it  should  I  pay  on  $6710 
worth  of  goods,  including  policy  at  $2.50? 

3.  At  4  ^  premium  and  policy  $1.75,  what  should 
I  pay  on  a  freight  of  furniture  worth  $2200  ? 

Art.  12.  The  term  Stocks  signifies  shares  in  incor- 
porated companies,  while  Bonds  represent  government 
and  municipal  securities,  and  mortgaged  securities  of 
corporations. 

1.  What  is  the  value  of  10  shares  of  railroad  stock, 
at  5  %  premium,  par  value  per  share  being  $50  ? 

2.  Bought  $1500  of  bonds  at  1.02  %,  what  did  I  pay 
for  them  ? 

3.  Sold  $500  in  railroad  stock  at  a  premium  of  5  ^, 
and  received  sugar  at  10  cents  per  pound,  how  many 
pounds  did  I  receive? 

4.  Purchased  50  shares  of  Pacific  railroad  stock  at  a 
discount  of  2^  %,  par  value  being  $100,  what  did  they 
cost  me? 

5.  Sold  $5000  in  bonds  at  a  premium  of  2J  ^,  what 
did  I  gain  after  paying  a  broker  \  %  for  selling  ? 

6.  Bought  $2600  of  canal  stock  at  3  %  below  par, 
and  sold  it  at  4  ^  above  par ;  how  much  did  I  pay, 
and  what  did  I  gain  by  both  transactions? 

Answers:  $26.61,  $170.25,  $89.75,  $525.00,  $1530.00, 
5250,  $4875.00,  $112.50,  $182.00,  $2522.00. 


64  PERCENTAGE. 


Art.  13.  Tax  is  a  sum  imposed  or  levied  upon 
society  to  defray  its  expenses. 

Art.  14.  Foil  tax  is  a  specific  sum  assessed  on 
male  citizens. 

Art.  15.  Duty  is  a  tax  levied  by  the  general  gov- 
ernment upon  imported  goods. 

Art.  16.  Specific  tax  is  a  fixed  sum  levied  upon 
specific  things  without  regard  to  value. 

Art.  17.  A  general  levy  is  an  assessment  upon 
projDerty  according  to  its  value. 

1.  The  general  levy  of  a  county  was  as  follows: 
8  m  ^^  for  school  fund,  4  m  ^  for  specific  purposes, 
Q  m  %  ^^^  sinking  fund ;  how  much  tax  will  I  have 
to  pay  on  $750  personal  property  and  on  $2800  real 
estate  ? 

Solution. — We  first  add  the  several  rates  together  to  ascer- 
tain the  whole  assessment,  viz  :  8 -[- 4 -|- 6  =  18  m  ^,  or  1.8. 
1^8_  c/^_  1750  _|_  $2800  r=  $3550  taxable  property.  lj«o  %  of 
$3550  z=z  $63.90. 

.  2.  Under  the  same  levy  what  tax  would  a  farmer 
have  to  pay  on  $256  personal  property  and  $8200 
real  estate,  including  poll  tax  for  three  male  persons 
@  $2  a  person,  and  a  special  tax  of  $5  for  a  piano 
and  $3  for  a  gold  watch  ? 

3.  What  would  be  the  duty  on  an  invoice  of  im- 
ported silks,  costing  $560  in  gold  (including  dutiable 
charges),  at  35  ^  less  10  %  ? 

Note. — The  85  cfo  is  to  be  reckoned  on  $560,  and  the  10  ^ 
to  be  taken  from  the  result.  It  will  be  observed  that  35  (fo  less 
10  (fo  is  not  25  ^,  but  31 J  ^. 

Answers:  $176.40,  $63.90,  $166.21. 

*  One  m.  per  cent,  is  one  mill  on  one  hundred  dollars. 


PERCENTAGE.  65 


Art.  18.  Marking  Goods. — This  is  done  by  select- 
ing samples  of  each  kind  or  quality  of  goods,  and 
putting  on  them  a  private  nfark,  indicating  the  cost 
price,  the  selling  price,  or  both. 

Every  house  has  got  its  own  peculiar  marks,  which 
generally  consist  of  the  letters  of  some  word  or 
phrase,  instead  of  figures.  For  instance,  if  we  take 
the  word  importance,  we  will  have  a  letter  for  every 
figure,  and  can  readily  substitute  the  former  for  the 
latter : 

1        2        3        4        5        6        7        g-i    9        0 

J        m       p         o         r         t        a         n        c         e 


When  a  figure  is  to  be  repeated,  an  additional  letter  is  need. 
Take  <7  in  this  case. 

The  selling  price  is  commonly  found  by  adding  to 
the  cost  price  a  certain  amount  per  cent,  to  cover 
the  freight  and  other  charges,  and  allow  a  remunera- 
tive profit. 

To  facilitate  labor,  some  merchants  make  this  per- 
centage an  even  part  of  100,  and  add  the  same  part 
of  the  cost  price  to  itself 

1.  To  add  12^  per  cent,  to  the  cost  price  of  goods 
at  $1.20  per  yard. 

12J  being  J  of  100,  add  J  of  120  (15c.)  to  itself,  which  will 
make  the  selling  price  %\.Zb. 

2.  To  add  25  %  to  books  at  S4.80  per  doz. 

25  being  J  of  100,  add  }  of  480  (120c.)  to  itself,  which  makes 
tlie  selling  price  %Q.0()  per  doz.,  or  50cts.  each. 

3.  To  30c.  add  20  %  profit. 

4.  $1.20       "       ^  %  charges  and  20%  profit. 

5.  1.75       "       2\%  freight    and  10%       " 

6.  Add  14  %  to  $1.75  7.  Add  15  %  to  $0.16 
8.        <^     12%  to    0.87  9.     "     25%  to    0.05 

10.        "     53^  to    1.67         11.     "     18^  to    3.16 

Answers:  $1.35,  $6.00,  $1.50,  $1.97,  $2.00,  $0.97, 
$0.18,  $2.56,  $0.06,  $0.36,  $371.00,  $3.73. 


66  PERCENTAGE. 


Art.   19.      MISCELLANEOUS  EXEKCISES. 

1.  Eeceived  a  consignment  of  tea,  which  I  sold  for 
S1678,  how  much  should  I  return  the  consignor,  after 
deducting  charges  $150,  and  commission  2^  %? 

2.  Sold  a  consignment  of  cloths  for  $6750,  with 
the  assistance  of  a  broker,  who  charged  me  ^  ^  ; 
what  amount  of  money  did  I  make,  com.  2|  %  ? 

3.  A  commission  merchant  sells  goods  for  his  prin- 
cipal to  the  amount  of  $3000,  and  charges  21^  com., 
what  does  he  make  by  the  operation,  after  Spaying  a 
broker  J^^  for  his  services  in  effecting  sales? 

4.  Insured  |  of  a  steamboat  worth  $18000,  at  1|  % 
premiura  :  what  did  it  cost  me  ? 

5.  How  much  should  an  insurance  company  pay 
to  an  insured  who  hold  a  policy  for  $2000  on  his 
dwelling,  the  damages  being  estimated  at  68%. 

6.  What  is  the  amount  of  loss  on  a  policy  insuring 
mdse.,  $1200;  fixtures,  $300;  building,  $4000;  dam- 
age on  building,  37-1-%;  mdse.  saved,  $300;  bal.  of 
mdse.  damaged,  75%;  loss  on  fixtures,  83J%. 

7.  A  merchant  holds  three  policies  of  insurance, 
as  follows:  iEtna,  $1000  on  leaf  tobacco,  $1500  on 
cigars,  $100  on  fixtures,  and  $200  on  retail  stock; 
Aurora,  $2000  on  leaf  tobacco;  American,  $1500  on 
leaf  tobacco  and  $1500  on  cigars.  Damage  on  leaf 
tobacco,  $3000;  cigars,  $2500;  retail  stock,  $300; 
fixtures,  $200.     What  must  each  company  pay? 

8.  After  reserving  5%  commission  on  sales,  amount- 
ing to  $520,75.  how  much  should  I  return  to  my 
principal  ? 

What  is  the  commission  on  the  following  amounts: 

9.  $364.15@3^%=         10.  $36.21@li%  = 

Answers:  $67.50,  $202.50,  $1360,  $2425,  $2216.67, 
$1333.33,  $2250,  $494.71,  $12.75,  $4.91,  $60.36,  $83.29, 
$0.54,  $5.80,  $135.49,  $4.30,  $1486.05,  $118.12,5. 


BILLS — INVOICES.  67 


XL    BILLS -INVOICES. 

WK/n  goods  are  sold,  it  is  the  duty  of  the  mer- 
chant, or  one  of  his  clerks,  to  make  out  a  statement 
of  the  quantity,  kind,  and  price,  of  each  article,  for 
the  satisfaction  of  the  purchaser,  and  to  enter  at  the 
foot  of  such  statement  the  whole  amount  of  the  pur- 
chase, with  the  payment  received,  if  any,  or  the 
terms  of  settlement.  If  the  goods  are  bought  to  sell 
again,  this  statement  is  commonly  called  an  Invoice^ 
otherwise  it  is  called  a  Bill^  especially  by  the  pur- 
chaser. 

A  bill  or  invoice  is  sometimes  delivered  to  the 
buyer  at  the  time  of  purchase,  but  it  is  usually  sent 
with  the  goods,  or  if  the  buyer  resides  at  a  distance, 
by  mail. 

An  invoice  should  specify  the  place  and  date  of 
sale,  the  names  of  buyer  and  seller,  a  description  of 
the  goods,  the  prices  of  boxes,  etc.,  used  for  packing, 
and  in  some  kinds  of  business,  the  terms  of  sale. 

When  goods  are  received,  the  quality  and  quantity 
are  compared  with  the  invoice,  and  the  selling  prices 
made  out  from  it,  after  which  it  is  filed  away  or 
pasted  in  a  book  for  future  reference. 

An  Account  is  a  statement  of  goods  sold  at  differ- 
ent periods  of  time.  Accounts  are  taken  from  the 
ledger,  and  often  contain  items  in  favor,  as  well  as 
against  the  buyer. 

Finding  the  cost  of  a  number  of  articles  at  a 
certain  price,  and  placing  the  amount  opposite,  is 
called,  in  bill-making,  extending;  adding  the  col- 
umns, footing  up. 

In  making  out  bills,  the  three  requisites  are 
rapidity,  legibility  and  accuracy.  The  principal 
is  accuracy. 


t)B  BILLS — INVOICES. 

Remabk. — The  bills  that,  come  in  are  usually  called  invoices 

\^ 

DRY  GOODS. 

New  Orleans,  March  4,  J  876. 
Mr.  W.  A.  Dickey, 

Bought  of  Charles  Shannon.  , 
26  pieces  Calico,  825  yds @  14c  %JlS^tS^ 

2  "  "       120     ''     @    9c       /a.  ?0 

12      "      Twilled  Muslin,  340  yds @  10c       ^  J  ^  0 

3  cases @  75c  \  'X^ 

Received  in  payment  his  note  at  90  days.  w  /A  J  ^^5^ 

Charles  Sni'tmoN,     * 
Per  H.  U. 

Cincinnati,  July  31,  1876. 
Mr.  R.  Nelson, 

Bought  of  Haseltine,  Macfarland  &  Co 

60     12     15  Braid  Bonnets, @$0.622  ^ 

68       6     Swiss  Straw  Bonnets, '.'    1.25 

70      4    7  Braid  "        *'   1.50 

80      2     7      "  **        "    3.00 

86      2    7"  "        "   3.75 

6    Pes.  No.  1  Tafft.  Ribbon, "      15 

5        "       "     2      "  "        "      28 

3       "      "    4      "  "        "      48 

2  "      "    6      "  "        "      75 

1  "       "  12       "  "        "    1.10 

3  "      Bonnet  Ribbon, "   2.00 

2  "            "             "  "    2.50 
3       1     Box  Ruches, !!!!!!.7.*.*.*.*.!.'.  "    1^50 

425       \       u  ((        ^         a   2.25 

210       \     Doz.  BunchesFlowersi.*.*!.*.'.'.'.*.*.'.'!!.'.'  ''  18.00 

I       "  "       Feathers, "36.00 

1     Po.  Black  Silk,  20  yds "      87^ 

S80.47  , 
Note. — The   numbers   in   the   column    on   the   left   are    those 
marked  on  the  boxes  and  packages. 


BILLS — INVOICES.  69 


y    GROCERIES  AND  LIQUORS. 

Cincinnati,  Sept.  1,  1875. 
Mr  James  O'Shaughnessy, 

Bought  of  King  &  Daly. 

I  Hhd.  N.  0.  Sugar,  ;ifg8  ^^1080  lbs @$0.07  % 

4  Bris.  jST.  0.  Molasses,  *|l  169  gals "  35 

1  Trs.  Rice, 7?§  630  lbs ;..  "  4 

20  Bags  Rio  Coffee, 3200  lbs "  11 

2  Half  chests  Black  Tea,  '50-1^72  lbs ''  25 

100-28 

3  "           "      Yng.  Hyson  do.   150  lbs "  50 

1  "            '      Imperial        do.     60  lbs "  40 

2  "          **      Gunpowder   do.   110  lbs ''  60 

1     "           "      Colong  Blk.  Tea,    45  lbs "  40 

6  Doz.  ground  Cinnamon, *'  40 

6     '*            "      Allspice, "  40 

6     "            ''       Pepper, "  40 

4  "            "      Mustard, "  75 

1  Box  5  lump  Tobacco,  i|J  108  lbs "  25 

1     "     pound  lump  do.     ^\l  124  lbs "  20 

1     "     Va.  pound     do.     ^|g  120  lbs "  35 

1     "     Slump           do.     i|gl25  1bs "  22 

20  Brls.  Rect.  Whisky, 800  gals "  17 

4     "      Ginger  Wine, 160  gals "  60 

i  Cask  French  Brandy, 40  gals "  4.00 

I      "     Port  Wine, 45  gals "  2.00 

10  Brls.  Bourbon  Whisky, 405  gals "  1.00 

\  Brl.  Holland  Gin, 20  gals...  .  "  1.50 


$1761.45 


*   Gross  Weight — weight  of  hogshead,  etc.,  and  contents. 
t  Tare  or  weight   of  bag,  box,  etc.     Ten  per  cent,  is  usually 
deducted  for  sugar. 

I  Gallons  in  each  barrel. 

**  Net    Weight — weight  of   goods  In   hogsheads,    etc. 


70  BILLS — INVOICES. 

Messrs.  Gaff  &  Baldwin,  Cincinnati,  June  8,  1876. 

Bought  of  Straight,  Deming  &  Co 

100  Boxes  cheese,  ^^oo  3590, @  $     .08  $ 

30  Firkins  Butter,  336o  2820, "         .15 

100  Boxes*  $20.00  Starch,  4810, ''         .05 

100      "      $25.00  Star  Candles,  4000,...  "         .20 

20  bbls.^  $25.00  Lard  Oil,  810  gals., ''         .85 

50     "    Mess  Pork, "      16.00 

10  Tierces  S.  C.  C.  Hams,  335o  300O, "         .11 

30  Kegs  Lard,  ijjg  1334, ''         .12J 

15  bbls.  Mess  Beef, "     15.00 

Com.  for  purchasing,  $1521.75, "     2^  fo 

Drayages, 16.00 

Insurance  on  $5000.00, 59.88 

QUEENSWARE.  $4152.87 
Philadelphia,  May  17,  1858. 
Mr.  W.  Anderson,                            To  Samuel  Asburj  &  Co. 
W.  A.  [c]    23  Crates  Queensware,  per  ship 
140  @  163.       Lancaster,  as  per  invoice  ren- 
dered,  £115  I65.  4d^bU.U 

Exchange  10  fo  prem.,  51.41 

Ins.  2J  %  @  5^  per  £,  14.33 

$579.88 
Int.  47  days,  from  Mar.  31  t^ 

May  17,  1858, 3.76 

Cash, $588.64 

Duties,  etc.,  on  the  above, 

Invoice,    £115 16s    4d 

Com.  2^fo     2  llsUd    £118  145  Sd 

@  $4.84  per  £,  is  $574.57,  duties  24  fo  $137.90 

Custom,  House  Fees, 1.00 

Freight,  £10  17  3  @  $4.80, 52.14 

Drayage, 9.50 

Cash, $200.54 

•  Price  of  empty  boxes  and  barrels. 


BILLS — INVOICES.  71 

Cincinnati,  December  1,  1876. 
Mr.  Newton  Thompson,  Germantown,  Ohio, 

Bought  of  William  Andersoa 

WA  [C]  3  doz.  Edged  Plates, @$0.40  $ 

115  10     "  "  "     "      50 

5     "         CC        "     "      50 

4     "  "    Dishes,  ea.  $1.75,  2.25, 

$2.75.t 
1     "  "     Bakers,  ea.  $L50,  2.00, 

$2.50. 

1  "  "    Beaded  Nappies,  each 

$1.75,  2.25. 
i     "  "     Chambers, "  3.50 

2  "  ''    Bowls, "     80 

24  "  ''  "     '<     60 

3  "  '*  ''     ''     50 

"  "     Pitchers, "  3.50 

"  colored       ''      ea.  $2.50,  4.00. 

4  "  ''     Bowls, *'     87^ 

5  "  "         "       "     65 

6  "  "         "       "     55 

9  Sets  CCTeas, "     20 

36     "     Painted  Teas, ''      20 

Crate, "  1.00 


3,  CA'PS^i 


$56.73 
HATS,  CA^S,^AND  FURS. 


Messrs.  D.  W.  Fairchild  &  Co.,         t^       u^    c  r.^      ^  l  r^  -it, 

'         Bought  of  Frost  &  Griffia 

tl328   ) 

1336  }   3c.    18  doz.  2  x  B.  C.  Shanghai,  @  $13.00 

1337  J 

{^|3    I    2c.      6     "     Fr.  felt  Hungarian,    "     21.00      . 

1491     Ic.      6     ''^  B.  C, ''     18.00 

6  cases  75  cents  each,  cooperage 

12^  cents  each,  drayage  37-^  cts., 5.63 

$473.63 

♦  "  CC  "  signifies  cream  color.     "  Teas,     cups  and  saucers, 
t  Three  qualities.  J  Numbers  on  cases. 


72  BILLS — INVOICES. 

Cincinnati,  Sept.  16,  1876. 
Messrs.  Lewis  Evans  k  Co., 

Bought  of  D.  W.  Fairchild. 

^  doz.  Men's  black  cass.  Hungarian,  @  21.00  $10.50 

1      u  a  u  u  li  u  27.00 

51     "        "        «'        «     "  33.00 

i     "        "        "        "     ''  24.00 

J     "         *'         "  broad  brim  wool,   "  14.00 

I       "         "         "   wove  Senate, "  12.00 

i    "         "         "   cashmerette, "  15.00 

^     "   Boy's       "   wool  Hungarian,.    "      7.00 

J     "         *'   caps  assorted, "  12.00 

i     "        "      "  "      "  9.75 

i     ''         "     cloth  caps,.    "  9.00 

i     "     children's  fancy  caps, "      8.00 

i     "  "  ''  "     "  13.00 

j\j  "     men's  cloth  caps, *'  14.00 

j\  "    boy's      "        "     "  10.00 

1     Case  at  75  and  1  at  50  cents,....  $264.63 


BOOKS. 

Philadelphia,  Oct.  9,  1876. 

Messrs.  Applegate  &  Co.,       Bought  of  Childs  &  Peterson. 

300  Kane's  Arctic  Explorations,  cloth,  $4.22 
100       "  "  "  sh.,      5.07 

50       "  ''  "     hlf.  antique  7.19 

50       "  "  ^        "     full      "        8.45 

50  Bouvier's  Law  Dictionary,  2  vols.,      8.45 
30        "  Institutes,  4     ''        12.67 

400  Shepherd's  Constitutional  Text  Book,     .63 

18  Cases  and  drayage, 24.17 

$3633.77 

Exchange  1  ^, $36.33 

Insurance, 44.50 

Freight, 85.41  

$3800,01 


BILLS — INVOICES. 


73 


HAKDWARE. 

Cincinnati, 
Messrs.  A.  C.  Morris  &  Co., 

Bought  of  Tyler 
6  Mouse  hole  Anvils 

^1.  ].  13.     0.  3.  22.     0.  2.  12. 
1.  2.  12.     1.  3.  16.     1.  1.  14. 

873  lbs.  @  141 
1  Case  best  cast  steel,  assorted,.  IX5 
liXi        4.  0.  26.         474  lbs.  @  17^ 

Pocket  Knives:  , 

Nos.    1212     1518    22    29    32      37 
Doz.        3  3      12      3       6        8 

Price,  4  s.     4s.  6d.   8s.  4s.9dl0s.  3s.9d. 

£11  5s.  9d.  @  $5.00 
5  doz.  narrow  Butts,  each,  3   in.    .85 

"      3^  "$1.15  10.00 
Less  10  fo  discount, 1.00 

9.00 
Less  Extra  12  ^, 1.08 

Package  and  Dray  age, 


June  3,  1876. 
Davidson  &  Co 


40 
82.95 


$207.35 


56.44 


Mr.  James  Brov^rn, 


Cincinnati, 


7.92 
3.25 
67.61 

$274.96 
July  3,  1876. 


10  Kegs  lOd.  nails,. 
6  "  '^' 
5 
2 
4 
3 
6 


8d. 

6d.      "     

4d.      'V   

8d.  fence  nails, 

8d.  brads, 

lOd.  finishing  nails, 

Amount  forwarded 


Bought  of  Guiou  &  Kizer 

@$3.50 

3.75 
4.25 
4.75 
3.75 
3.75 
4.50 

$141.50 


Long  weight,  cwL  qrs. 


74  BILLS — INVOICES. 

Amount  brought  forward, 1141.50 

4  Kegs  8d  nails, @  $4.75 

2  Doz.  No.  1  tinned  bottom  coffee  mills,   "    12.00 

o     "        ''     2       ''  "  "         "        "      5  00 

3  u  u      3         u  u  a  u  u       I'j-^Q 

2  "     Britannia  hoppers, "  7.00 

4  "     Brass                '*        "  4.00 

6  "     Iron                   "         "  3.00 

4  "     No.  8  X  Janus  faced  locks, "  13.00 

2  ''       ''     ^'  a      "          ''          "     "  15.00 


$335.00 


TAILORING. 

Richmond,  Va.,  Feb.  3,  1876. 
Mr.  Michael  Tracey, 

To  James  Humble. 

Sep.  9,  For  making  and  trimming  one  overcoat, $12.00 

"     1  silk  velvet  vest, 10.00 

"     ^  doz.  pair  socks, @  $2.37^        1.19 

Feb.  5    1876.        Settled,  ^23.19 

James  Humble. 


FARMING. 

Lexington,  Ky.,  Oct.  29,  1876. 
Mr.  W.  R.  Henderson, 

Bought  of  R.  S.  Wharton. 

Jan.     3,  3  Durham  heifers, @  $25.00 

1  Bay  2  yr.  colt, "      50.00 

Apr.  11,  140^  Bushels  corn, "         .50 

5^  Doz.  chickens, "        ^'^^h 

Aug.    9,  123^  lbs.  butter, "         .18| 


Cr.  $225.97 

Apr.    4,  By  25  Hogs,  3147  lbs.  @    $0.3^ 

"       IPlow, $15.00. 

Oct.  30,    "     Cash  for  balance, 

R.  S.  Wharton. 


BILLS — INVOICES.  75 


~^^^ 


EXERCISES    FOR   PEN    AND    PAPER. 

The  pupil  should  provide  himself  with  bill  paper, 
and  make  out  bills  from  the  following  transactions, 
using   his  own  name   as  clerk  or  principal,    as  he 
refers. 

1.  Apr.  3,  1876,  Sold  to  Mrs.  E.  Nelson,  22  yds. 
black  silk  @  $1.25,12  yds.  black  silk  velvet  @  $4.87, 
15  yds.  linen  @  75  cts.,'47  yds.  W.  flannel  @  621  cts., 
to  be  charged  to  account  of  Eichard  Nelson. 

/  2.  Sold  to  Mr.  H.  Schnicke,  1  overcoat  @  $17.50, 
^  doz.  shirts  @  $32.50,  .J  doz.  pocket  hkfs.  @  $5.75, 
^  doz.  pairs  socks  @  $3.75.  Eeceived  paj-ruent  in 
cash,  A.  B.  (clerk).     Feb.  3,  1876. 

7^.  June  9,  1876.     Sold  to  Cyrus  Wright  on  order 

/of  A.  J.  Eice,  2  doz.  Gillot's  pens  @   12^  cts,  1  box 

F.  envelopes  @  $1.50,   .}   doz.  penholders  @  50  cts., 

1  copy  Bjn'on  @  $1.50,  2  copies  Shakespeare's  2)lays 

@  $2.50.     Indorsed  on  order. 

Note. — This  transfiction  is  a  very  common  one,  and  should  be 
thoroughly  understood  by  the  learner.  It  supposes  us  to  be  in- 
debted to  A.  J.  Rice,  who,  in  his  turn,  is  indebted  to  C.  Wright, 
in  whose  favor,  he  (Rice)  draws  the  following  order  on  us: 

"  Cincinnati,  June  8,  1876. 
Messrs.  Nelson,   Kizer  &    Co.    will    please    let  C. 
Wright  or  bearer liave  goods  to  the  amount  of  Twenty 
dollars,  and  charge  to  ray  account. 
$20.00  A.  J.  EicE.'^ 

The  amount  of  the  bill  being  less  than  that  of  the 
ordur,  Wright  is  permitted  to  keep  the  latter,  after 
we  write  across  the  back  of  it,  "  June  9,  paid  $8.50, 
N.,  K.  &Co.^' 


76  BILLS — INVOICES. 


The  amount,  $8.50,  is  then  charged  on  our  books 
to  Eice. 

A  better  way  is,  take  Eice's  order,  charge  him  with 
the  $20.00,  and  give  Wright  a  due  bill  for  the 
balance,  $11.50. 

4.  May  13,  1876.  Sold  to  H.  J.  Minor,  Louisville, 
3  chests  Congou  tea,  marked  H.  J.  M.— 21,  22,  23. 
No.  21,  102  lbs.  gross,  tare  21  lbs.  ;  No.  22,  103  lbs. 
gross,  tare  22  lbs. ;  No.  23,  99  lbs.  gross,  tare  20  lbs., 
@  75  cts. 

5.  Sept  9,  1876.  Sold  to  Eobert  O'Brien  125  yds. 
carpet  @  $1.12,  .10  pieces  Irish  linen,  198  yards  @ 

*26  cts.,  6  pieces  muslin,  71  yds.  @  12^  cts.,  5  pieces 
French  merino,  175  yds.  @  87  cts.,  I  doz.  silk  um- 
brellas @  $51.50, 12  pieces  black  silk  velvet,  250  yds., 
@  $3.25.     Paid  drayage  $1.00,  and  insurance  @  1^%. 

6.  Jan.  9,  1876.  Sold  to  Andrew  Spence,  Pittsburgh, 
6  hhds.  sugar,  and  shipped  same  on  steamboat  Bos- 
tona,  Miles,  master.  The  hhds.  were  marked  and 
numbered  ''A.  S.,  5,  6,  7,  8,  9,  10.''  No.  5  weighed 
1424  lbs.,  tare  27  lbs.  ;  No.  6,  1573  lbs.,  tare  31  lbs. ; 
No.  7,  1397  lbs.,  tare  35  lbs.;  No.  8,  1576  lbs.,  tare 37 
lbs. ;  No.  9,  1498  lbs.,  tare  30  lbs. ;  No.  10,  1675  lbs., 
tare  36  lbs.  @  12^  cts.  Paid  drayage,  $2.50,  insur- 
ance 1^  %. 

7.  Dec.  3,  1876.  Sold  to  Mrs.  Sophia  Dodd,  20  yds. 
I  muslin  @  15  cts.,  18  yds.  French  merino  @  87^  cts., 
1  silk  bonnet  @  $5.  Eec'd  in  payment  B.  E.  Gooley's 
due  bill,  W.  Dodd's  favor  for  $30,  and  gave  our  due 
bill  for  balance  of  note  unpaid. 


BILLS— INVOICES.  77 


EXERCISES  IN  MAKING  EXTENSIONS. 

The  following  exercises  can  all  be  solved  by  the 
short  methods  explained  in  another  part  of  this 
work:     They  are  designed  for  the  blackboard. 

67i@ 
157  '' 
216  '' 
917i@ 
119f  " 
175  '' 
143J  ^'  . 
216f  " 
116J  '' 
718^  ^' 

1.  Find  the  cost  of  15  yards  muslin  @  12^  cents, 
22  yards  silk  @  821  cents,  45  yards  ticking  @  25 
cents,  150  yards  satfn  @  $2.25,  45  yards  calico  @  8^ 
cents,  125  yards  M.  Delaine  @  27  cents,  121  yards 
French  merino  @  62^  cents,  6  pieces  sheetings,  197 
yards  @  33J  cents,  12  pieces  shirtings,  376^  yards,  @ 
12^  cents,  15  gross  spools,  $2.25. 

2.  Find  the  cost  of  2  cases  assorted  cassi meres, 
175  yards,  @  $1.75,  8  pieces  blue  cloths,  216  yards, 

*917J  at  a  dollar  =$917.60 

$917.50  1 800  @  5.17=  $4136.00 

6  2  @,  6.17  deducted      10.34 


;0.83i  4197  @ 

$1.25 

6191 

@$3.20 

0.121 

464*  '^ 

7.621 

116/:. 

''  3.37^ 

1.87^ 

119f  ^' 

1.45 

26^ 

"  1.82^ 

6.55* 

3671® 

1.35 

197 

@  3.13 

1.50 

4881  ^' 

1.65 

682# 

"  1.25 

2.17 

771J-  ^' 

2.121-  -j-798 

"  5.17 

6.55 

1671  " 

2.50 

677^ 

*'  4.87^ 

1.75 

719f  " 

1.63 

35 

'^  0.35 

1.37i 

711i  " 

1.35 

115i 

"  3.121 
^'  0.18| 

0.45 

125  ^' 

0.65 

417 

6505.00  $4125.66 

458.75  =J  of  the  cost  at  a  dollar. 
45.87  J==-j-^  of  the  cost  at  60  cts. 

6009.62J 


78  -LONG    DIVISION. 


XII.  LONG  DIVISION. 

Art.  1.  The  previous  operations  in  division  have 
been  performed  mentally^  the  learner  writing  only  the 
quotient.  This  is  called  Short  Division,  and  is  to  be 
preferred  when  the  divisor  is  a  small  number,  or 
caa  be  reduced  to  a  small  number,  as  200,  12000, 
which  by  pointing  off  the  ciphers  are  reduced  to  2 
and  12  (Art.  9,  p.  41.)  But  when  the  divisor  is  19, 
23,  79,  536,  etc.,  the  operation  would  be  too  difficult 
and  tedious  to  perform  mentally.  In  such  case  the 
greater  part  of  the  process  has  to  be  written^  and  is 
known  by  the  name  o?  Long  Division. 

Example  1.— To  divide  3147  by  6. 

SHORT    method. 

6)3147 
~524| 
long  method. 

dividend,  quotient. 

6)3147(524^  Explanations. — 1.  To  perform  this  operation 

OA  ^      we  say,  6  in  31,  6  times,  and  write  5  in  the  quo- 

tient  and  multiply  it  on  6,  which  makes  30.    This 

14  we  write  under  the  31. 

'i  2  2.  We  now  subtract  the  30  from  the  31  as  we 

would  perform  an  operation  in  subtraction.    The 

27  remainder    is    1.      Instead  of   supposing  this    1 

oj.  to  stand  before  the  4  in  the  dividend,  we  bring 

down  the  4  to  it,  which  makes  14. 

3  3.  Six  in  14,  2  times,  we  write  2  in  the  quo- 

tient,  multiply  it  by  6   and  write   the   product 
underneath. 

4.  Subtracting  this  12  from  the  14  we  find  a  remainder  of  2,  to 
which  we  annex  7,  brought  down  from  the  dividend,  making  the 
number  27 

5.  Six  in  27,  4  times,  we  write  4  in  the  quotient,  multiply  it  by 
6,  and  write  the  product  24  underneath. 

6.  Subtracting  as  before  we  find  a  remainder  of  3,  under  which 
we  write  the  divisor  6,  making  |. 


LONG   DIVISION.  ,  79 

2.  To  divide  834716  by  723. 
723)  834716(1154||| 

723  '  This  is  the  product  of  1X723. 

2  Tiie  remainder  after  subtracting  723 

'1117  from  834,  with  7  brought  down. 

»   723  3  The  product  of  1X723. 


4  The  remainder   of  1117— 723,  and  1 
*  3941  brought  down 

^  3615  5  Tlie  product  of  723X5. 

-- — -^  6  The  remainder  of  3941—3615,  and  6 

3266  brought  down. 

'  2892  "^  The  product  of  723X4. 

-^ 8  The  last  remainder=||| 

Note. — The  pupil  should  put  a  dot  under  each  figure  brought 
down,  to  prevent  its  being  taken  twice. 

3.  To  divide  67314968  by  163000. 

163|000)67314|968(412i|fgg| 
652   * 


211 
163 

"484 
326 

158 

Remarks. — 1.  Instead  of  using  the  whole  divisor  in  finding  a 
quotient  figure,  it  will  generally  do  to  use  only  the  first  one  or 
two  figures.  For  instance,  in  Ex.  2,  say  7  in  8,  instead  of  723 
in  834;   and  in   Ex.  3,  say  16  in  67,  instead  of  163  in  673. 

2.  The  products  should  never  exceed  the  numbers  above  them; 
(number  3  should  not  exceed  number  2;)  if  they  do,  a  smaller 
number  should  be  put  in  the  quotient. 

3.  For  every  figure  brought  down  from  the  dividend,  there  should 
be  one  in  the  quotient.     Where  the  divisor  is  not  contained  in  the 

small  dividends,  as  the  310  in   the  68   (Art.  2,  Ex.  1),  a  cipher 
should  be  written  in  the  quotient,  and  another  figure  taken  down. 

4.  The  divisor  can  not  be  contained  more  than  9  times  in  the 
new  dividends. 

*  These  three  figures  are  a  part  of  the  remainder,  as  shown 
in  the  quotient. 


80 


LONG   DIVISION. 


Art.  2.  To  divide  dollars  and  cents,  we  reduce 
them  to  cents,  then  our  quotient  or  answer  will  be 
cents,  which  are  easily  converted  into  dollars,  by  in- 
serting the  decimal  jDoint. 

1.  To  divide  $3168.20  by  310. 

310)316820(1022  or  $10.22 
310 

"682 
620 

~620 
620 

2.  Divide  9765837  ''    65.  Ans.   150243||. 

3.     1763-^76=  4.     3167-^-119= 

7964--87='-  71438-^-320  = 

89737--98  =  67898^764= 

77168-^19=  78637^-892= 


Total  remainders,  138 

5.*  $10000--7109  = 

7185--1990= 

67416---  144r= 

3784--  642= 


Total  remainders,  958 

6.     $140.98-^-671  = 

730.45~-126  = 

164.87-^-144= 

1710.14-f-166  = 


Total  quotients,  $479.06         Total  quotients,  $17.44 

PRINCIPLES  OF  DIVISION. 

Art.  3.  If  we  divide  the  price  of  a  number  of 
things  of  equal  value  by  the  number,  we  obtain  the 
price  of  one. 

Art.  4.  The  quotient  will  always  be  in  the  same 
name  with  the  dividend  or  number  to  be  divided. 
If  the  dividend  be  dollars,  the  quotient  will  be  dol- 
lars ;  if  it  be  rods,  the  quotient  will  be  rods. 

*  Reduce  these  to  cents  before  dividing:  $10000=1000000  cents 
CSee   Note   page    38,)    «ind  omit  the  remainders. 


EXERCISES    IN    MULTIPLICATION    AND    DIVISION.     81 


EXERCISES  IN  MULTIPLICATION  AND  DIVISION. 

1.  If  23  yds.  of  muslin  cost  S3. 45,  what  will  one 
yard  cost?* 

2.  If  117  men  can  do  a  piece  of  work  in  48  days, 
how  long  will  it  take  three  times  that  number  to  do 
it? 

3.  How  many  men  can  do  a  piece  of  work  in  5 
days,  that  took  10  men  25  days? 

4.  If  a  case  hold  29  pieces  of  muslin,  how  many 
will  it  take  to  hold  7250  pieces  ? 

5.  If  15  men  can  do  a  certain  piece  of  work  in  75 
days,  how  long  will  it  take  1  man  to  do  it  ? 

6.  If  7  dozen  silver  spoons  cost  $35.35,  what  will 
3  dozen  cost? 

Note. — Find  the  cost  of  1  dozen,  then  the  cost  of  3. 

7.  If  two-sevenths  of  a  ship  cost  $14602,  what  will 
the  seven-sevenths,  or  the  whole  ship  cost? 

METHODS  OF  PROOF. 

Art.  5.  Division  and  Multiplication  being  con- 
verse operations,  the  one  is' proved  by  the  other. 

DIVISION.  PROOF. 

88)3715(97  97=quotient. 

342  38==divisor. 


295  776 

266  291 

"29  rem.     3686+  the  rem.  29=3715=dividend 

MULTIPLICATION.  PROOF. 

multiplier,  prodact.  multiplicand. 

465  25)11625(465 

25  100 


2325  '      162 

930  150 

11625  l25 

125 
6  


82  TIME. 


XIII.  TIME. 

TO  RECKON  TIME. 

Art.  1.  Business  men  usually  reckon  30  days  to 
the  month;  but  when  a  note  is  given  at  one,  two  or 
three  months,  it  falls  due  on  the  same  day  of  the  month 
it  was  given,  plus  the  days  of  grace. 

Some  notes  and  bonds  draw  interest  from  date. 
When  such  is  the  case,  the  time  is  computed  as  fol- 
lows : 

1.  What  is  the  difference  of  time  between  January 
3,  1878,  and  February  9,  1879  ? 

3rrs.  mos.     days. 

Operation.      1879         2         9 
1878         1         3 


Am.  116 


ExPiiANATiON. — We  call  January  the  first  month,  February 
the  second,  etc. 

2.  What  is  the  difference  in  time  between  April  3, 
1878,  and  January  1,  1879? 

yrs.  mos.      days. 

Operation.      1879         1         1 
1878        4        3 


8      28 

Find  the  difference  of  time   between  the  following 
dates : 

January  1,  1874,  and  April  2,  1876  = 
October  9,  1871,  and  Jan.  1,  1875  = 
June  23,  1875,  and  Dec.  9,  1878  = 


Total,  8  yrs.  11  mo.  9  ds. 


SIMPLE   INTEREST.  83 

XI  \.  SIMPLE  INTEREST. 

Art.  1.  Interest  is  a  percentage  charged  for  the 
use  of  capital.  It  is  regulated  by  the  year  or  month. 
6  per  cent,  (per  annum)  signifies*  szo:  dollars  on  every 
hundred  dollars  for  a  year.* 

Interest  may  be  divided  into  simple  and  compound. 
Simple  Interest  is  percentage  on  capital  alone. 
Compound  Interest  is    interest  reckoned    on    both 
capital  and  interest. 

TERMS. 

The  terms  are  Principal,  Rate,  time  and  Amount. 
The  Principal  is  the  sum  or  capital  loaned. 
The  Bate  is  the  percentage  charged. 
The  Amount  is  the  sum  of  principal  and  interest. 

Note. — The  legal  interest  of  the  United  States  is  6  per  cent. 
Vi  hen  no  per  cent,  is  named  in  this  book,  6  per  cent,  is  understood. 
2.  Mills  are  omitted  in  the  answers. 

Art.  2.  The  interest  on  any  sum  of  dollars  for  60 
days,  is  equal  to  as  many  cents  as  there  are  dollars.f 

The  int.  on  $100  for  60  days  is  100  cents  or  $1.00. 
u      u    1250    "     ''       "         1250     •'       ^'   12.50. 
Find  the  interest  on  the  following: 
1.  $1749  for  60  ds.  2.  $1009  for  60  ds. 

785    ^'    '^     "  719   '^    "     '' 

9000    "    "     ^'  5000    '^    "     " 


Total,    $115.34  Total,  $67.28 

*  It  is  customary  to  reckon  interest  for  all  rates  at  6  per  cent., 
and  afterward  to  increase  or  diminish  as  necessary.     See  Art.  6. 

t  Since  the  interest  on  $100  for  360  days  is  $6  (Art.  2),  for  60 
days,  it  is  one-sixth  as  much  or  $1.00;  but  $1  is  100  cents  or  as 
many  cents  as  there  are  dollars  in  the  princioal. 


S4  SIMPLE   INTEREST. 


Art.  3.  To  find  the  interest  for  any  number  of  daySy 
we  take  that  part  of  the  interest  at  60  days,  that  the 
number  of  days  is  of  60. 

To  find  the  interest  of  $120  for  30,  20,  15,  12,  or 
10  days. 

The  interest  on  $120  for  60  days,  is  $1.20. 
For  30  days  it  is  J  of  $1.20,  or  60  cents. 
For  20  days  it  is  |  of  1.20,  or  40  cents. 
For  15  days  it  is  i  of  1.20,  or  30  cents. 
For  12  days  it  is  |  of  1.20,  or  24  cents. 
For  10  days  it  is  I  of    1.20,  or  20  cents. 

Reason. — Since  30  days  is  J  of  60,  the  interest  for  30  days  will 
be  J  of  that  for  60  days;  20,  15,  12,  and  10,  are  also  equal  parts 
of  60. 

Note. — When  the  days  are  not  even  parts  of  60,  we  divide 
them  into  even  parts.  For  18,  we  take  16  and  8 ;  for  27,  take  12 
and  16;  for  37,  take  30,  6,  and  2;  for  110,  take  60,  30,  and  20. 

2.  Find  the  interest  on  $211  for  93  days. 

2   11    =int.  for  60  days.  The    student,     after    some 

1     1   055:=   a       a    30      "  practice,  should  not  lose  time 

f          -j  QP- (t       t<       q      n  by   writing   the    divisors,    or 

2^   lUO—                 ^  ^j^g  jj^^,g  ^^  ^Yie  right,  as  in 

«3    27   =s   *'       "    93      *'  *^^^  example. 


TABLE. 

ALIQUOT    OR    EVEN    PARTS    OF    SIXTY    DATS. 

To    be  committed  to   Memory   by  the   Pupil. 
30  days  =i      12  days=  I      5  days=y'^     2  days=3'^ 

90      '<       i^        10      "      '        4.      '^      1        1       *'       J 

ip;      u      — 1  f\      a      —  1        ^      '*      —  1 

10  — :j  D  — yu  ^  — TT(5 

ds.  ds. 

3.    $797.00  for  10=$1. 33         4.    $1000  for  2= 

*  Ititerest  is  seldom  reckoned  on  cents.     If  less  than  60,  reject 
them,  otherwise  add  a  dollar  to  the  dollars. 


Simple  interest. 


85 


$1000 

71 

61 

190 


days. 
,00  for  27= 
.97  for  47= 
.80  for  45= 
.27  for  16  = 


6. 


days. 

^1799.14  for  98= 

387.66  for  67  = 

199.44  for  41  = 

450.22  for  29= 


Total,    $6.03 


Total,  $35.75 


days. 
7.  $719.99  for  11  = 
55.18  for    9  = 

88.17  for  69= 
466.00  for  78= 


$1997.00  for 

7.88  for 
17.97  for 


days. 
13= 
54= 
35  = 


10.00  for  120= 


Total,  $8.47 

days. 
$1000.00  for  97  = 
650.00  for  67  = 
10.70  for  13= 
127.57  for  51  = 
368.17  for  118= 
718.57  for  125= 


Total,  $4.71 


10. 


days. 

$1999.20  for  23= 

361.74  for  18= 

78.93  for  23  = 

1467.20  for  34= 

7100.18  for  77- 

29.00  for  99  = 


Total,  $46.76 


Total,  $108.96 


Art.  4.  To  find  the  interest  for  years  and  months. 
In  a  year  there  are  6  sixty  days ;  therefore  we  mul- 
tiply the  interest  for  60  days  by  six  times  the  num- 
ber of  years,  and  as  there  are  half  as  many  sixty 
days  as  months,  we  multiply  the  interest  for  60 
days  by  half  the  number  of  months. 

Kecapitulation. — Consider  the  dollars  cents ^  and 
multiply  by  6  times  the  number  of  years  plus  half  the 
number  of  months j  and  for  the  days  take  aliquot  parts  as 
before. 

1.  To  find  the  interest  of  $120  for  1  year  4  months 
and  20  days. 


86 


SIMPLE   INTEREST. 


1.20 

Explanation. — The  interest  for  60  days  is  120  cents;  S 

for  1  year  and  4  months  it  is  8  times  120  or  960  cents;  

and  for  20  days  it  is  J  of  120,  or  40  cents,  making  the  9.60 

sum  $10.00 — the  interest  required,  40 

Arts.     $To7oO 

2.  Find  the  interest  of  ^240  for  3  years  4  months 
and  10  days.  Ans.  $48,40. 

3.  What  is  the  interest  of  $1467.45  for  2  years  6 
months  and  17  days.  Ans.  $224.21. 

Find  the  interest  of  the  following: 


yrs. 

mos. 

days. 

4.  $321.00  for 

2 

3 

15.*  $44.14 

1767.00  for 

7 

4 

21.     783.66 

897.25  for 

3   i 

6 

27.     192.41 

898.57  for 

2     - 

7   ^ 

25.t  148.09 

716.27  for 

2 

1 

9=   90. .57 

810.98  for 

1 

6 

7=  73.94 

50.00  for 

3 

7    . 

18=   10.90 

8.00  for 

9 

•■^./ 

27         4.48 

yrs. 

mo» 

days. 

5.  $3140.79  for 

^  y 

7 

7  = 

795.17  for 

2 

1 

1  = 

3.90  for 

3 

5 

15  = 

1057.57  for 

1 

11 

11  = 

"tCotal,  $526.01 

yrs. 

mos. 

dfivs. 

6.  $2674.57  for 

1 

8 

21  = 

7143.45  for 

2 

1 

18  = 

1742.67  for 

1 

9 

13=      - 

2100.00  for 

2 

1 

1  = 

4109.85  for 

1 

6 

17  - 

Total,  $2022.35 

*Find  the  interest  for  2  years  4  months,  and  deduct  the  int.  foi 
16 'days. 

t  Call  this  2  years  8  months,  and  deduct  the  int  for  5  days. 


SIMPLE   INTEREST.  87 


yrs.  mos.  days. 

7.  $7856.00  for         1  1  29  = 

677.19  for         3  3  3= 

287.17  for         1  7  16  = 

97.19  for         5  10  14= 

10.10  for         1  3  19= 


$743.95 

yrs. 

mos. 

davs. 

$57.87  for 

2 

6 

14= 

120.14  for 

7 

7 

7  = 

340.00  for 

9 

1 

24= 

1657.00  for 

1 

3 

24= 

769.75  for 

2 

3 

18= 

$487.40 

Art.  5.  Having  the  interest  at  6  per  cent,  to  find  the 
\rMrest  at  any  other  rate. 

This  is  done  by  taking  aliquot  parts  of  6,  and  in- 
creasing or  diminishing  the  interest,  as  the  rate  is 
more  or  less  than  6  per  cent. 

At  2  per  cent,  the  interest  is  I  of  that  at  6  per  cent. 

At  4    "       "       it  is  I  less  than  at  6  per  cent. 

At  8  "  ^'  it  is  J  more  than  at  6  per  cent. 
.  At  10  "  ''  it  is  ig«  of  that  at  6%  ;  so  we  have 
only  to  move  the  decimal  point  in  the  Q%  interest 
one  phice  to  the  right,  and  divide  by  6.  For  15  %^ 
we  move  the  decimal  point  in  the  same  way,  and  di- 
vide by  4 ;  and  for  20  %  by  3. 

1.  Let  the  interest  at  6  %  be  $240. 

At    2  %  it  will  be  I  as  much,  or  $  80 

"      8  %   "     "     "    I  more,  or  320 

"    10  %   "     '*     ''  10  times  i  of  $240,  or  400 


88  SIMPLE   INTEREST. 


2.  To  find  14  %,  7i  %,  8J  %,  and  10|  %  of  $350  for 
60  days. 

4)3.50   int.  at  6    %  ^^^f^  3  ^f;-  f^^  \    ^^ 

.87,5"     "    \\%  '.14  5    ''     "     i?^ 

84723~~  "     "   71  % 

$3.50     int.  at "6    %  $3.50    int.  at  6  ^ 

1.16.6  '^      "  2    %  1.75      ''      "  3    % 

»19.4  "      *'  J  %  .87  5  "      "^H  % 

$4.86                 8J%  $6.13              10^% 

yrs.  mos.       days. 

3.  $798.18  for      6  1         6  @  9%  =$438.10 
1000.00  for      4  2         4  @  7%=  292.44 

yrs.  mos.       days. 

4.  $340  for          2  2        20  @  2i%  = 

600  for          3  4        15  @  6i%  = 

850  for          1  2        12@8|%  = 


Total,  $237.22 


5.  Find  the  interest  of 

mos.         days. 


$617.18  for  3         18  @  15%  = 

460.74  for  2  5  @  18%  = 

765.12  for  8         16®  20%  = 


Total,  $151.55 

6.  Find  the  interest  on  the  following  at  10% 

$710  for  92  days=  7.  $496  for    91  day8  = 

1978  for  27      "    =  671  for    86     "     = 

8889  for  128     "    =  100  for  104     ''     = 

75  for  117     "    =  269  for    73     "     = 


Total,  $351.47  Total,  $36.91 

•Art.  6.   It  is  customary  for  bankers  to  lend  money^ 

and  discount  b}    the  month  instead  of   by  tne  ytar. 


SIMPLE   INTEREST.  89 

This  percentage  is  easily  converted  into  6%   interest, 
and  the  work  performed  with  as  much  ease  as  before : 

1  %  per  month  is  12%  per  year,  or  2  times  6% 
\\%    "        "      is  18%    ^'  ^  ''     or  3      "     6% 

2  %    "        "      is  24%    ''      ''     or  4      "     6% 

1.  Find  the  interest  of  the  following : 

$65.00  for  80  days® 2  %  per  month  = 
40.00    ''   33     ''    @\\%     ''       "      = 
190.00   **   63     "    @2  %     "       *'      =   • 
700.00   ''93     "    @3  %    *'       "      = 


Total,   $77.20 

Art.  5.  The  work,  when  computing  interest,  can 
often  be  abbreviated.  Sometimes  advantage  may  be 
taken  of  the  aliquots  of  hundred  ;  at  other  times  it  will 
be  of  advantage  to  transpose  the  terms  and  consider  the 
days  as  dollars  and  the  dollars  as  days;  or  the  rate 
(if  it  is  some  other  rate  than  6%)  may  be  reduced 
mentally  to  6%.  For  instance,  in  the  second  question 
in  the  last  group,  the  $40  may  be  considered  $120,  and 
transposing  the  term  and  the  33  multiplied  by  2,  mak- 
ing 66c  the  answer. 

It  will  materially  abridge  the  operation  and  expedite 
the  labor,  if  the  learner  will  observe  to  avoid  the  use 
of  all  lines,  figures  or  marks  that  are  not  absolutely 
necessary.  As,  for  instance,  when  using  aliquot  parts, 
to  write  only  the  results  of  division,  as  shown  in  the 
following  example: 

Interest  of  $321  for  2  years,  1  month  and        8.21 
22  days  at  10%  per  annum.  "40^25 

Explanation. — Mentally  it  is  found  that  there  are         1.07 
Vl\  60  days  in  2  years  and  1  month,  to  multiply  by  ^07 

which  we  divide  by  8.     The  division  by  6  and  the      

multiplication  by  10  were  performed  simultaneously,       41.302 
giving  $68,836  or  $68.84  as  the  answer.  ~7:r7:7?. 

68.806 


90  COMPOUND    INTEREST. 


XY.    COMPOUND  INTERJEST. 

Art.  1.     In  Compound  Interest  the  interest  is  con 
verted   into   principal    ev^ery  quarter,    half  year  or 
year.     Capital  is  thus  more  rapidly  increased,  than 
by  simple  interest. 

Any  person  acquainted  with  the  principles  of 
simple  interest  will  readily  understand  how  to  com- 
pute compound  interest. 

1.  What  is  the  compound  interest  of  $1000  for  1^ 
years  at  6  %,  payable  semi-annually  (half-yearly)? 

The  interest  of  SIOOO  for  6  mo8.=  S30.00 
Add  the  principal,  1000.00 

Amount  for  6  mos.  $103oToO 

Interest  on  $1030  for  6  m^os.,  30.90 

Amount  for  1  year,  $1060.90 

Interest  on  $1060.90  for  6  mos.,         31.827 


Amount  for  18  mos.,  $1092.727 

Principal,  1000.00 

Compound  interest  for  1-J  years,      $92.73 

2.  Find  the  compound  interest  and  amount  of  $1865 
for  3  years,  3  months,  at  8  ^,  payable  tri-^monthl}^. 

3.  What  is  the  compound  interest  and  amount  of 
$486  for  4  years,  at  10  ^,  payable  annually? 

4.  What  is  the  compound  interest  and  amount  of 
$672  for  4  years,  at  6  %  per  annum  ? 

Answers:  $1092.73,  $848.38,  $2412.56,  $711.55, 
$92.73,  $176.38,  $547.56,  $225.55. 

Kemark. — At  6  per  cent,  money  will  double  itself  in  11  years, 
10  months  and  21  days.  At  5  per  cent.,  in  14  years,  2  months, 
and  15  days.  At  3  per  cent.,  in  23  years,  5  months,  and  lOJ 
days. 


ANNUAL  INTEREST.  91 


XVI.    ANNUAL  INTEREST. 

Annual  Interest  is  the  term  applied  to  interest 
on  a  note  that  is  drawn  with  the  clause  ^'  interest 
payable  annually."  When  this  interest  is  not  paid 
at  the  end  of  the  year,  it  draws  simple  interest  till 
paid. 

1.  A  note  for  $300  at  3  years,  6%  interest,  pay- 
able annually,  had  nothing  paid  on  it  at  maturity. 
How  much  was  due? 

Int.  on  $300  at  6%  =$18  =  int.  for  1  year. 
3 


$54  =  int.  for  3  years. 
Int.  on  $18  for  2  years,  2.16 
"  '^  1  year,    1.08 

Principal,       300.00 

$357.24  amount. 

2.  What  is  the  amount  of  a  note,  at  the  end  of  4 
years  for  $368,  for  2  years,  8%  interest,  payable 
annually,  that  had  nothing  paid  on  it  until  settle- 
ment? 

Note. — When  a  note  is  overdue  interest  is  calculated  up  to 
the  date  of  maturity,  as  in  Ex.  1,  and  simple  interest  is  calcu- 
lated on  the  amount  from  maturity  till  paid. 

3.  What  should  I  pay  at  maturity  to  redeem  my 
note  for  $800,  payable  3  years  after  date  with  10% 
interest,  payable  annually,  nothing  having  been 
paid  at  maturity? 

4.  A  note  for  $720,  at  4  years,  6%  interest,  pay- 
able annuall}^,  had  nothing  paid  at  maturity.  How 
much  was  due? 

5.  What  is  the  amount  of  a  note,  at  the  end  of  3 
years,  for  $1268,  payable  2  years  after  date,  bearing 
S%  interest,  payable  annually,  no  payments  having 
been  made  until  settlement? 

Answers :  $497.91,  $1064,  $738.84,  $1597.32,  $357.24, 
$908.35,  $497.92. 


92  PARTIAL   PAYMENTS. 


XVII.    PARTIAL  PAYMENTS. 

Art.  1.  Notes,  bonds,  etc.,  drawing  interest,  are 
sometimes  paid  by  installments,  and  the  amounts 
thus  paid,  indorsed  on  them.  The  legal  rule  for 
computing  interest  on  installments,  may  be  expressed 
thus : 

Apply  the  payment  to  the  discharge  of  the  inter- 
est, and  if  there  is  a  remainder,  subtract  it  from  the 
debt.  When  the  payment  is  less  than  the  interest 
due,  it  is  not  applied  to  the  discharge  of  the  interest 
or  debt,  but  is  indorsed  on  the  note  until  the  install- 
ments exceed  the  interest;  then  the  sum  of  the 
payments  are  computed  as  below. 

1.     $576.  Cincinnati,  Oct.  9,  1875^ 

On  demand,  1  promise  to  pay  Eobert  Ingles, 
or  order,  five  hundred  and  seventy-six  dollars,  with 
interest,  value  rec'd. 

Samuel  Dunning. 

On  the  note  are  the  following  indorsements; 

RecM  Dec.  16,  1875,  $100 
"  Feb.  28,  1876,  3 
'^      July  27,      "        150 

Required  the  amount  due  Sept.  3,  1878. 

yi  mos.    ds. 

From     1875     12     16 
Take      1875     10       9 


Difference,  2       7,  or  67  days. 

Amount  of  note,  $576.00 

Interest  on  $576  for  67  days,        6.43 

Total  amount  due,  $582.43 


PARTIAL  PAYMENTS.  93 

Total  amount  due,  $582.43 

Installment,  to  be  subtracted,    100.00 

Balance  due,  $482.43 

The  second  payment  is  less  than  the  interest  due, 
and  no  calculation  is  required. 

From  Dec.   16th,  1875,  to  July  27th,  1876,  is  7 
mos.,  11  days. 

Balance  due,  $482.43 

Interest  for  7  mos.  11  days,    17.75 

Amount  due,  500.18 

Amount  of  payments,  153.00 

Balance  due,  $347.18 

From  July  27  to  Sept.  3d,  is  38  days. 

Balance,  347.18 

Interest  for  38  days,  2.19 

Amount  due  Sept.  3,  1876,  $349.37 

2.  $650.  Boston,  June  3,  1868. 

For  value  rec'd,  I  promise  to  pay  on  demand 
t  •  H.  Crooks,  or  order,  six  hundred  and  fifty  dollars, 
with  interest  at  6  %  per  annum. 

Indorsements.  J  .    i  .    JJAVIS. 

Jan.  6,  1870,  $95 
Oct.  13,  1870,  350 
Jun.  3,  1875,     12 

Sept.   7,  1877,  paid  the  balance,  how  m.uch  was  it? 

Ans.  $405.92. 

3.  On  a  note  drawn  Sept.  3,  1877,  for  $650  with 
legal  interest,  there  are  the  following  indorsements  : 

Oct.     4,        $100 
Nov.    3,  2 

Dec.  19,  210 

Apr.     3,  1878.  the  balance  ;  how  much  was  it? 

Ans.  $354.32. 


M  BANK   DISCOUNT. 


XVIII.  BANK  DISCOUNT. 

Art.  1.  Discounting  notes  consists  in  buying  them 
at  less  than  their  nominal  value,  or  the  amount  for 
which  they  are  drawn.  The  difference  between  the 
nominal  value  and  the  price  paid  is  called  discount. 

There  are  two  kinds  of  discount :  True  Discount, 
which  is  interest  paid  in  advance  on  the  present 
value  of  a  note,  and  Bank  Discount,  which  is  interest 
paid  in  advance  on  the  face  of  the  note.  The  latter 
resembles  compound  interest,  as  it  is  interest  on 
both  interest  and  principal.* 

When  a  note  is  discounted  in  bank,  the  interest 
of  the  note  for  the  time  it  has  to  run,  and  at  the 
banker's  rates,  is  deducted  from  the  sum  called  for 
by  the  note.  This  species  of  discount  is  therefore 
reckoned  in  the  same  way  as  interest. 

1.  How  much  discount  should  be  deducted  from  a 
note  of  $500  at  90  day^  2 

$5.00=  int.  for  60  days. 
2.50=  "       ''    30     ^' 
.25=  '^       "      3     ''     (grace) 

Ans.  $7.75 

2.  $1500,  Columbus,  Jan.  8,  1879. 

Sixty  days  after  date,  I  promise  to  pay  Messrs. 
M'Ewen  and  Banfill  one  thousand  ^ve  hundred  dol- 
lars, value  received. 

William  Dodd. 
Eequired  the  discount  at  6  ^.  Ans.  $15.75. 

*  The  present  worth  of  a  note  drawn  for  $100,  payable  in  a 
year  at  6  per  cent.,  is  $94.34,  and  the  interest  is  $5.66;  that  is, 
the  principal  and  interest  together,  are  equal  to  $100,  or  the 
face  of  the  note;  so  when  a  banker  discounts  from  the  face  of 
a  note,  he  discounts  on  both  principal  and  interest. 


BANK   DISCOUNT.  95 


Bankers  prefer  lending  money  on  short  time,  and 
by  the  day,  instead  of  by  the  month.  Notes  are 
usually  drawn  for  30,  60,  or  90  days  ;  and  interest 
18  always  charged  on  the  days  of  grace, 

1.  What  is  the  bank  discount  on  a  note  of  $120 
at  60  days,  at  ^  %  per  month?  Ans.  $1.26. 

2.  Find  the  discount  on  a  note  of  $575.75  at  90 
days,  at  the  same  rate.  Ans.  $8.92. 

3.  What  is  the  bank  discount  on  a  note  of  $450 
for  60  days  at  2  %  per  month?  Ans.  $18.90. 

Remark. — The  discount  on  $450  at  2  per  cent,  per  month,  is 
the  same  as  the  discount  of  4  times  $450,  qf  $1800  at  6  per  cent 
per  annum. 

4.  How  much  money  should  be  paid  by  a  banker 
who  discounts  a  note  of  $350  at  30  days,  at  1^  % 
per  month?-  Ans.  $344.22. 

5.  What  will  be  the  proceeds  of  a  note  drawn  for 
$670  at  60  days,  at  2  %  per  month?     Ans.  $641.86. 

6.  At  1^  %  per  month,  how  much  proceeds  should 
be  recovered  on  a  note  of  $1749.57,  drawn  at  90 
days?  Ans.  $1668.12. 

7.  Find  the  discount  on  a  note  of  $1678.25,  drawn 
at  90  days  at  IJ  %  per  month  ? 

8.  At  2i  %  per  month,  what  is  the  discount  on  a 
note  of  $688  at  90  days? 

9.  At  If  %  per  month,  what  will  be  the  proceeds 
of  a  note  drawn  for  $6784,  at  60  days? 

Answers  to  the  foregoing:  $47.99,  $65.03,  $6534.69. 

10.  Find  the  discount  on  the  following: 
$1310.00  for  60  daj^s  @  2     %  per  mo. 

746.87  "  90  "   *'  1-J  "  ^'   *< 


219.56  "  30  *'   "  1 
1867.25  '^  20  "   "  2^  "  ** 
1367.00  ^'  15  "   "  3  "  " 


$152.57 


96 


BANK  DISCOUNT. 


.  $1673  for  30  days 

(a>, 

1 

6789 

''     3  nios. 

u 

2 

1987 

"     9  mos. 

a 

1 

6745 

"  10  days 

a 

n 

%  per  mo. 


$693.21 


Find  the  amount  of  proceeds  of  the  following: 
12.  $3768  for  10  days  @  4     %  per  mo. 
1767     "    15     "       "  8     ''      ''      " 
8767     ^'     6     "       '*  1^  "      "      '^ 


$14165.43 


13.  $167.39 
978.00 
897.87 


for  2  mos. 

U        Q        U 

u     3     (( 


@  ^  %  per  mo. 
''  20  ''  "  an. 
<<  25      '^     "      " 


$1900.25 


Note. — As  a  review  exercise  the  pupil  might  cal- 
culate interest  on  cents  as  well  as  dollars. 

Art.  2.  Bankers  frequently  discount  notes  that  are 
partly  matured  ;  when  such  is  the  case,  the  following 
table  will  assist  the  accountant  in  computing  the  dis- 
count: 

A    TABLE 

Showing  tho  number  of  days  from  any  day  in  one  month,  to  the  same  day  in  any 
other  month,  throughout  the  year. 


MONTHS. 

i 

X5 

u 
a 

ft 
< 

^ 
S 

>> 

ft 

1 

> 

o 

1 

January, 

365 
334 
30fi 
275 
245 
214 
184 
153 
122 
92 
61 
31 

31 
365 
337 
306 
276 
245 
215 
184 
153 
123 
92 
62 

59 
28 
365 
334 
304 
273 
243 
212 
181 
151 
120 
90 

90 
59 
31 
3'i5 
335 
304 
274 
243 
212 
182 
151 
121 

120 

89 

61 

30 

365 

334 

304 

273 

242 

212 

181 

151 

151 

120 

92 

61 

31 

365 

335 

304 

273 

243 

212 

182 

181 

150 

122 

91 

61 

30 

365 

334 

303 

273 

242 

212 

212 

181 

153 

122 

92 

61 

31 

365 

334 

304 

273 

243 

243 

212 

184 

163 

123 

92 

62 

31 

365 

335 

304 

♦274 

273 
242 
214 
183 
153 
122 
92 
61 
30 
365 
334 
304 

304 

273 

245 

214 

184 

153 

123 

92 

61 

31 

365 

335 

334 

February, 

^m 

March, 

275 

April, 

?-W 

May, 

■;ii4 

183 

July, 

153 

August, ..-. 

}n 

September, 

91 

October, 

61 

November, 

30 

December,...: 

365 

BANK   DISCOUNT.  97 


Use  of  the  Table.^To  find  the  time  from  Feb.  13 
to  March  23,  in  the  following  example  :  In  the  left 
hand  column  we  find  February  in  the  second  line, 
and  running  the  eye  along  till  we  come  under 
"  March,'^  we  find  the  number  28;  hence  from  Feb. 
13  to  March  13  is  28  days  ;  to  March  23d,  will  there- 
fore be  10  days  more,  or  38  days.  The  discount 
will  be  reckoned  for  93  days  minus  38  days,  equal  55 
days. 

1.  A  note  drawn  on  Feb.  13,  1878,  for  $900,  at  90 
days,  was  discounted  on  March  the  23d,  at  2  ^  per 
month,  how  much  was  paid  by  the  borrower? 

Ans.  $867.  ' 

2.  What  proceeds  should  be  paid  on  a  note  of 
$346  at  90  days,  drawn  on  Nov.  3d  and  discounted 
on  Dec.  the  7th,  at  IJ  %  per  month?  ^riS.  $335.79. 

3.  A  note  of  $689,  made  Sept.  9,  payable- in  60 
days,  was  discounted  on  Oct.  5th,  at  2  %  per  month, 
what  was  the  discount?  Ans.  $16.99. 

Note. — If  the  decimals  be  carried  out  to  three  or  four  places, 
the  cents  may  differ  slightly  from  the  following  totals. 

(4.) 

Face  of  Date.  Time.         When  Rate  of  Disc't. 

Note.       •  Disc'td. 

$167.50  Jan.    3,  1879  60  days  Feb.    7,  2    ^  per  mo. 

9876.00  Feb.    7,     "  90     "     Mar.  12,  2^  fo    "      " 

789.00  Jun.  18,     '*  30     "     July    3,4%    "      " 

1897.00  Feb.  21,     "  90     "    Apr.    1,  1^  fc    '' 


Total,     $555.24 
(5.) 
1676.37  Apr.    3, 1879  90  days  May    9,   2    %  per  mo. 
679.39  Mar.    9,    "      30     "    Apr.   3,   2^  fo    "      " 
7168.00  Jun.  13,    "      60     "     July    9,    1^  fo    ''      " 

816.37  Aug,  12,    "      30     "     Sep.    6,   21  fo    "      " 

Total,     $167.74 
7 


98  BANK  DISCOUNT. 


(6.) 

*No^te°.^  Date.  Time.       ^isc'td.         Rate  of  Disc't.       Discount 

2676.00  Jan.  9,  1879  90  days  Feb.  1,  li  fo  per  mo. 

7187.00  Feb.  3,  "   60  "  Mar.  13,  \\<fo    "  " 
768.21  Mar.  6,  **   30  "•     Apr.  3,  2  ^  "   " 

314.00  Apr.    7,    "      90     "     May  15,   2    ^    "      " 

Total,      $181.98 

7.  A  note  of  $1675  drawn  on  IS'ov.  3,  at  3  months, 
was  discounted  on  Dec.  2,  at  1^  %,  what  was  the 
discount? 

8.  What  amount  of  proceeds  will  arise  from  dis- 
counting a  six  months'  note,  drawn  for  S197,  at  2  ^ 
per  month  ? 

9. 

Am't.  of       J.  .  ^.  When  dis-  B-^te  of  proceeds 

Note.         ^**®-  ^*"*®*  counted.  Discount  iToceeas 

$6785  Dec.    6, 1878,  6  mo.  Dec.  29, 1878,  2  %  per  mo. 
3748  Jan.    3,1880,5   "     Feb.    3,1880,2^%    "      " 
6983  Mar.    9,    "      4   "     Jun.    8,    "      \\^o 


Total,   $16277.22 
10. 
$3784  May    6, 1880,  2  mo.  July    3, 1880,  2  fo  per  mo. 
6987  Jun.    8,    *'       3   "     Aug.  27,    "       \\%    "     " 

7854  July  24,    "      4  "     Sep.  17,    "       1   %    "     " 

Total,  $18371.58 

11.  What  amount  of  money  should  I  receive  on  a 
note  of  $675,  discounted  at  35  days  (having  35  days  to 
run,)  1^  %  per  month  ? 

12  June  3d,  discounted  my  note  of  $350  at  10  %, 
having  30  days  to  run,  required  the  discount? 

13.  Feb.  6,  1878,  had  A.  Seers'  note  of  $500,  dated 
20th  Dec,  1877,  discounted  at  \\  %  per  month,  time 
to  run  33  days,  what  were  the  proceeds? 


TRUE    DISCOUNT.  99 


XIX.    TRUE  DISCOUNT. 

Art.  1.  True  Discount  is  the  difference  between 
the  present  worth  of  a  note  and  the  amount  for  which 
it  is  drawn. 

The  present  worth  of  a  note  or  bill  due  at  a  future 
time  without  interest,  is  such  a  sum  as  would,  if  put 
at  interest  for  the  same  time  and  rateper  cent.,  amount 
to  the  debt;  and  the  difference  between  this  sum  and 
the  debt  is  the  discount. 

1.  What  is  the  true  discount  on  a  note  of  $700  for 
90  days  at  6  %  ? 

The  amoftntof  a  dollar  for  93  days  is  $1.0155,  by 
which,  if  we  divide  $700,  we  will  find  the  present 
worth. 

Operation,  $1.0155)700.0000(689.31     . 
60930 
Proof.— The  interest  on  $689.31  ^THv^T^ 

92  days,  is  $10.68,  or  $10.69  nearly,  ^UTUU 

which,  if  added   to  the  principal,  81240 

will  give  $700.  -^— 

Note.— The  interest  of  $1  for  91395 

90  days  is   .0165.     The    present  ■ 

value  of  $1.0155  for  93  days    is,  32050 

therefore,  $1,  and  accordingly  the  30465 

present  value  of  $700  for  93  days  " 

is  $700.00,  divided  by  $1.0155  or  15850 

$689.31,    and  the  discount    $700,  10155 

$689.31,    or  $10.69.  

2.  What  is  the  true  discount  on  a  note  of  $575  for 
90  days  at  6  ^  ? 


3.  What  is  the  true  discount  on  a  note  for  $880  for 
120  days  at  8  ^  ? 

Answers:  $23:39,  $11.55,  $10.69^ ^ 

*  Four  ciphers  are  annexed  to  the  ^700  to  correspond  with  the  divisor; 
the  quotient  from  this  will  be  dollars ;  by  annexing  two  more  ciphers,  the 
answer  will  appear  in  cents. 


100  DISCOUNTING  INT.- BEARING   NOTES. 


XX.    DISCOUNTING  INT.-BEARING  NOTES. 

Art.  1.  There  are  three  methods  of  reckoning 
discount  on  interest-bearing  notes  :  Bankers'  Method^ 
Brokers'  Method^  and  Equitable  Method. 

Art.  2.  Bankers  calculate  interest  on  the  face  of 
an  interest-bearing  iiote  up  to  the  date  of  maturity 
(including  days  of  grace),  and  discount  the  amount 
(principal  and  interest)  for  the  unexpired  time. 

Art.  3.  Money  brokers  reckon  interest  on  the 
face  of  an  interest-bearing  note  up  to  the  date  of 
discount,  and  discount  on  the  face  of  the  note  from 
the  date  of  discount  to  the  date  of  maturity,  at  the 
difference  between  the  rate  of  interest  and  rate  of 
discount. 

Art.  4.  By  the  Equitable  method,  interest  is  reck- 
oned on  the  face  of  the  note  up  to  maturity,  and 
true  interest  is  calculated  on  the  amount. 

1.  What  amount  of  proceeds  will  arise  from  a  note 
dated  January  3,  1875,  for  $1200,  payable  2  years 
after  date,  with  6%  per  annum  discounted  April  18, 
1876  @  10%? 

Bankers'  Method. 

$12.00 
12 


144.00  int.  2  yrs. 
.60  int.  3  days 


144.60  int.  @  6%  for  2  yrs.  3  ds. 
1200.00  jDrincipal 


$1344.60  amount 

The  unexpired  time  from  date  of  discount,  April 
18,  1876,  to  maturity,  January  6,  1877,  is  264  days. 


DISCOUNTING   INT. -BEARING   NOTES. 


101 


$13.44  60 

4 

53.78  4  int.  for  240  ds. 
5.37  8  int.  for  24  ds. 


59.16  int.  for  264  ds.@6%  per  an. 


19.72 

19.72    "      "      " 

98.60    "      ''      " 

1344.60  amount 
98.60  discount 


$1246.00  proceeds 


Brokers'  Method. 


"2% 
10% 


$12.00 
07 

1876  4 
1875  1 

18 
3 

84.00  int. 
6.00  " 
3.00  '' 

1  yr.  2  mos.        1  3 
1  mo. 
15  ds. 

15 

$93.00  " 

1  yr.  3  mos.  15  ds.  @  6%  per 

an. 

$12.00 
4.00 

12 

1877  1 
1876  4 

18 

48.00  int. 

3.00  " 

.60  '' 

8  mos.  6% 
15  ds.  6% 
3  ds. 

51.60  '' 
17.20  " 

8  mos.  18  ds.  6% 

8  "   18  "  2% 

$34.40  " 

8  "  18  "  4% 

102  DISCOUNTING   INT. -BEARING   NOTES. 


Interest  in  favor  of  borrower,      $93.00 
Discount  against  "  34.40 

Diiference  in  favor  of     "  58.60 

Principal,  1200.00 

Proceeds,  $1258.60 


Equitable  Method. 

The  interest  the  same  as  by  the  Bankers'  method : 
$144.60;  amount,  $1344.60. 

The  amount  of  $1  for  264  days  (the  unexpired 
time)  at  10%  per  annum  is  $1.0716* 

$1.0716')  1344.60 
Proceeds,  $1254.68 

It  will  be  noticed  that  the  Bankers'  method  is 
more  favorable  to  the  lender ;  the  Equitable  method, 
to  the  borrower. 

2.  What  is  the  present  value  (Jan.  6,  1876)  of  a 
note,  dated  March  18,  1875,  for  $268.50,  payable  3 
years  after  date,  with  S%  interest  per  annum;  rate 
of  discount,  12%  per  annum? 

Note. — These    examples    should    be   worked    by   all   three 

methods. 

3.  What  amount  of  money  should  I  receive  on  a 
note,  dated  May  8,  1875,  for  $668.35,  payable  2 
years  after  date,  with  6%  interest  per  annum;  dis- 
counted Sept.  11,  1875;  discount,  8%  per  annum? 

Answers:  $243.84,  $262.57,  $263.42,  $647.66,  $680, 
$659.93,  $68,  $261.97,  $263.34,  $659.85. 


COMPLEX   PERCENTAGE.  103 


XXI.    COMPLEX  PERCENTAGE. 

Art.  1.  Complex  Percentage  embraces  all  those 
calculations  of  percentage  the  result  of  which  can  not 
be  ascertained  by  the  process  explained  under  Simple 
Percentage. 

Art.  2.  To  find  the  gain  per  cent,  when  the 
actual  gain,  the  principal,  and  amount  are  known. 

1.  What  is  the  gain  per  cent,  on  goods  bought  at 
$2.50  and  sold  at  $2.75? 

SOLUTION :     $2.75  selling  price. 
2.50  cost  price. 

25  actual  gain. 

The  actual  gain  on  the  investment  or  cost  price  ($2.50)  is  25 
cents.  To  ascertain  the  gain  per  cent.,  i.  e.,  the  gain  on  the 
100  cents,  the  first  step  will  be  to  find  the  gain  on  one  cent — 
viz:  if  250  cents  bring  25  cents,  one  cent  will  gain  as  much  as 
250  is  contained  in  25,  viz:  y^^  cent ;  if  we  gain  ^^  cent  on  one 
cent,  we  gain  on  100  cents,  100  times  y^^,  viz:  10  cents;  i.  e.,  10  4, 

25X100      ,.  ^ 
formula:    JLILjOi y_^iQ  g^ 


In  the  formula  we  multiply  the  actual  gain  by  100  and  divide 
the  product  by  the  cost  price,  the  result  gives  us  the  gain  per 
cent. 

Note. — This,  like  nearly  all  examples  in  complex  percent- 
age may  be  worked  by  proportion,  which  would  read:  The 
cost  price  (250)  is  to  100  as  the  gain  on  the  cost  price  is  to 
the  gain  on  100  or  the  gain  per  cent. 

2.  What  is  the  gain  per  cent,  on  goods  bought  at 
$1.60  and  sold  for  $1.96? 

3.  Bought  one  bbl.  flour  for  $8.25  and  sold  it  for 
$8.91.     What  was  my  gain  per  cent.? 

Answers  arranged  promiscuously  on  page  118. 


104  COMPLEX  PERCENTAGE. 

Art.  3.  To  find  the  Principal,  when  the  rate  and 
amount  are  known. 

1.  How  much  of  $2448  can  I  invest  in  grain  after 
retaining  2  ^  commission  for  buying? 

Solution. — The  amount  $2448  includes  the  cost  of  the  grain 
(principal)  and  the  commission.  The  commission  is  to  be  paid 
on  the  principal  and  not  on  the  amount,  and  as  the  principal 
is  unknown  we  can  not  ascertain  the  commission  hy  reckoning 
simple  percentage.  Every  principal  is  100  per  cent,  of  itself, 
plus  2  ojo  commission  makes  the  amount  equal  102  per  cent. 
If  $2448^:102  ojo,  and  the  principal  =  100  %,  we  find  the 
principal  by  multiplying  the  amount  by  100  and  dividing  the 
result  by  102,  viz: 

102)1244800 

2400  the  principal. 

proof:     $2400  the  principal. 

2  %  rate  of  commission. 

4800 
2400 


$2448.00  the  amount  received  for  investment. 

Note. — The  same  result  may  be  ascertained  by  fractions, 
thus:  The  amount  is  equal  to  the  principal  and  commission. 
The  commission  is  2  ^  or  yj^  of  the  principal  The  principal 
is  ^gg,  and  the  amount  is  as  much  as  the  principal  and  commis- 
sion, viz:  |gj  or  jj.  Hence,  we  have  to  divide  $2448  by  |^, 
thus: 

51    ^^    50 
$2558  -^— =  $2448  X— =-  $2400. 

50      ^^^    ^; 

2.  Allowing  2\  ^f)  commission  on  a  sale  of  300 
bbls.  of  flour  for  $2567,60,  how  much  can  I  invest  in 
sugar  after  reserving  2  %  commission  for  buying  ? 

Note. — The  commission  for  selling  must  be  ascertained  by 
simple  percentage.  By  deducting  the  same  from  the  proceeds 
($2567.60),  we  obtain  the  amount.  The  remainder  of  the  ex- 
ample can  be  solved  by  either  of  the  foregoing  solutions. 


COMPLEX  PERCENTAGE.  105 

3.  What  would  be  the  face  of  a  draft  at  a  discount 
of  IJ  (fo  to  cover  investment  of  $858.92? 

Note. — In  this  example  the  amount  is  smaller  than  the 
principal,  The  discount  is  to  be  taken  from  the  principal — $1 
of  the  principal  contains  \\  cents  discount  and  98 J  cents 
face.  Hence,  98J  cents,  or  98.5,  is  to  be  divided  into  $858.92 
X 100  to  ascertain  the  principal. 

4.  At  a  premium  of  2  ^  what  should  be  the  face 
of  a  draft  to  cover  an  investment  of  $749.70  ? 

Art.  4.  To  find  the  Eate  when  the  principal  and 
the  gain  or  loss  are  known. 

1.  What  is  the  rate  per  cent,  of  a  dividend  of  $42 
on  an  investment  of  $600  ? 

Solution. — To  obtain  the  rate  per  cent,  of  dividend  means 
to  ascertain  the  amount  of  dividend  on  the  hundred.  If  $600 
bring  $42,  $100  (being  \  of  600)  will  bring  \  of  $42,  viz:  $7 
on  $100,  which  is  7  cL 

7 

formula:    M2<_Mi^.$7  or  7  ^. 


2.  Eeceived  a  dividend  of  $63  on  an  investment  of 
> ;  what  was  the  rate  per  cent.  ? 

3.  With  an  investment  of  $4944,  what  rate  per 
cent,  will  bring  $618  ? 

Art.  5.  To  find  the  Eate  of  Income  for  a  given 
investment  obtained  at  a  discount  or  for  a  premium. 

1.  If  I  invest  in  6  ^  interest-bearing  bonds,  pay- 
ing 10  %  premium,  what  per  cent,  of  income  will  I 
receive  ? 

Solution. — If  I  buy  at  10  cfo  premium,  I  pay  $110  for  a  bond 
of  $100.     The  investment  of  $110  brings  me  but  $6— 

hence:     exlOp^jO^ 

lip  11         >'  /" 


106  COMPLEX  PERCENTAGE. 

2.  Bought  5-20  (6  %)  bonds  @  12  ^  premium ; 
whate  rate  per  cent,  income  will  I  receive,  including 
J  %  brokerage? 

Note. — The  premium  and  brokerage  added  will  give  what 
was  paid  above  the  par  value. 

2.  Bought  railroad  bonds  bearing  8  ^  interest  at 
a  discount  of  4  %  ;  what  will  be  the  rate  per  cent,  of 
my  income  ? 

Art.  6.  To  find  the  Cost  Per  Cent,  of  interest- 
bearing  stocks  in  order  to  make  a  certain  per  cent, 
income. 

1.  What  should  I  pay  on  the  dollar  of  a  6  ^  bear- 
ing bond  to  make  an  income  of  10  %  ? 

Solution.— A  $100  6  cfo  bond  will  pay  $6,  and  $6  is  10  ^  of 
$60.  Hence,  I  must  buy  6  ^  bearing  bonds  @  60  cents  on  the 
dollar  to  make  10  ^. 

FORMULA  :     —^ —  =  60. 

10 

2.  At  what  rate  should  I  buy  a  7^  ^  bearing  bond 
to  make  an  iacome  of  10  ^  on  my  investment? 

3.  How  much  should  I  pay  on  the  dollar  of  an  8  ^ 
bond  to  make  an  income  of  12  ^  ? 

Art.  7.  To  find  the  Cost  of  an  investment  bought 
at  a  premium  or  discount  and  sold  at  a  premium  or 
discount,  when  the  gain  and  rates  of  premium  and 
discount  are  known. 

1.  Bought  stock  at  4  %  discount  and  sold  it  at  6  % 
premium,  gaining  thereby  $200 ;  what  was  the  amount 
of  my  investment  ? 

Solution. — Buying  at  4  ^  discount  and  selling  at  6  ^  pre- 
mium gives  a  profit  of  10  9^. 

FORMULA :     ^^0X100^  $2000,  face  of  bonds. 
10 


COMPLEX  PERCENTAGE.  107 

4  9^  of  $2000=:  $80;  that  taken  from  the  par  value  gives 
$1920  as  the  investment. 

2.  Bought  stock  at  10  %  discount  and  sold  it  at  2^  ^ 
premium,  realizing  $87-^;  what  was  the  par  value  of 
the  stock,  and  how  much  did  I  invest  ? 

3.  Bought  stock  at  2  ^  premium  and  sold  it  at  2J 
%  discount,  thereby  losing  $450 ;  what  was  the  par 
value,  and  how  much  did  I  invest? 

Art.  8.  To  ascertain  the  Time  a  note  has  to  run, 
the  discount,  etc.,  being  known,  so  that  a  certain  rate 
will  be  equal  to  the  interest  at  another  certain  rate. 

1.  How  long  will  a  note,  discounted  at  20  %  per 
annum,  have  to  run  to  make  22J  %  interest  per  an- 
num. 

Solution. — The  discount  at  20  ^  per  annum  is  equal  to  the 
interest  @  25  9^  per  annum  (Proof:  Interest  on  $100  for  1  yr. 
@  20  ^  ^  $20,  giving  $80  as  the  proceeds,  25  ^  of  which  will 
be  $20).  Hence,  in  1  year,  or  360  days,  a  discount  of  20  %  will 
equal  the  interest  at  25  %.  To  ascertain  in  how  many  days  a 
discount  would  bring  22J  0/^  interest,  use  the  following — 

360  days  X  25%  X  2+      ,  .  , 

FORMULA :    ^—^ ^^—-^ — ^  which 

22^%X5 
8 

;^ 

stands  canceled:  ^^0  X  25  X  ^i^^OO  days. 

% 

Proof.— The  discount  for  200  days  on  $100(^^20^^  ---=$11.11, 
giving  as  proceeds  $88.89,  the  interest  on  which  for  200  days  @ 
22.]%  is  $11.11. 

The  numbers  used  in  the  formula  are  360,  the  number  of 
days  in  a  year;  25,  the  rate  of  fnterest  per  year;  22 J,  the  rate 
of  interest  for  the  time  desired;  2 J,  the  difference  between  the 
rate  of  discount  and  the  rate  of  interest  for  the  desired  time; 
and  5,  the  difference  between  the  rate  of  discount  per  year  and 
the  rate  of  interest  per  year. 

2.  How  long  will  a  note,  discounted  at  the  rate  of 


108  COMPLEX  PERCENTAGE. 

20  ^  per  annum,  have  to  run  to  make  24  %  interest 
per  annum? 

.  3.  To  make  11 J  ^  interest  per  annum,  how  long 
would  a  note,  discounted  at  10  ^  per  annum,  have  to 
run  ? 

Art.  9.  To  ascertain  the  Eate  of  Gain  on  articles 
which,  by  being  bought  at  a  certain  lower  rate,  will 
produce  a  certain  higher  rate  of  gain. 

1.  If  an  article  be  bought  at  10  ^  less,  and  the  rate 
of  gain  thereby  increased  15  %,  what  would  be  my 
rate  of  gain  ? 

Solution. — By  buying  at  10  cfo  less,  we  pay  90  cents  on  the 
dollar.  Hence,  by  buying  the  cost  is  90  cents  on  the  dollar. 
10  cents  gained  on  90  cents  is  a  gain  oflli^.  \i  \\\  cents 
gain  on  the  dollar  in  the  cost  will  make  15  cents  on  the  dollar 
in  the  gain,  the  entire  gain  per  cent.,  the  following  formula 
will  work  out  the  selling  price : 

15X100       15X9X100       ,^. 
formula:    — = — — — =135. 

11-^  100 

$1.35  being  the  amount  an  investment  of  $1.00  will  produce. 
Hence,  we  gain  35  cents  on  the  dollar,  or  35  %. 

Note. — The  example  may  be  worked  by  proportion,  thus: 
Hi  :  15 -i^  100  :  135.  For  \\\  is  to  15  as  100  is  to  135;  or,  the 
gain  per  cent,  buying,  is  to  the  gain  per  cent,  selling,  as  the  cost 
price,  is  to  the  selling  price. 

2.  If  an  article  cost  me  12 J  ^  less,  my  rate  of  gain 
was  increased  16  %]  what  was  my  rate  of  gain? 

3.  What  will  be  my  rate  of  gain,  if  I  buy  an  article  ' 
at  10  %  less,  and  thereby  gain  16f  %  more. 

Answers:  50  ^,  12  %,  36  ^  ;  400  days,  300  days, 
200  days ;  $10200,  $10000,  $700,  $630,  $1920  ;  66f  cts., 
73^,60  cts.;  Si%,H%,^^%;  121^,7^^,7%; 
S872,  $735,  $2454.32,  $2400;  24^^,  8  ^,  10^,22J%. 


COMPLEX  PERCENTAGE.  109 

Art.  10.  To  find  the  amount /or  icMch  a  note  may 
he  draivn  to  realize  a  certain  sum  after  being  discounted. 

1.  Kequired  the  face  of  a  90-day  note  which  will 
realize  $275.23,  after  being  discounted  at  2%  per  month. 

[nterest  on  $1  for  93  days  at  2%  per  month  =  .062. 

Proceeds  of  SI  --^  $1,000  —  .062  =  .938. 

Since  there  are  as  many  dollars  in  the  principal  as 
the  proceeds  of  $1  is  contained  times  in.  the  proceeds 
given,  $275,230-^.938  will  give  the  principal  required, 
$293.42  +. 

Proof.— Interest  on  $293.42  for  93  days  at  2%  per 
month  =^  18.19  -4-,  which,  subtracted  from  $293.42, 
leaves  $275.23,  the  proceeds. 

2.  The  proceeds  are  $212.60,  time  63  days,  rate  1^% 
per  month ;  required  the  principal.  ^ 

3.  What  principal  will  realize  $120  proceeds  in  6 
months  at  10%  per  annum? 

4.  The  time  is  three  months,  rate  10%  per  an.,  pro- 
ceeds $168.97;  what  is  the  principal? 

5.  The  rate  is  12%  per  annum,  proceeds  $693.75, 
time  4  months;  required  the  principal. 

Art.  11.  To  find  the  rate  per  cent.,  when  the  prin- 
cipal, interest,  and  time  are  given. 

1.  The  principal  is  $300,  time  60  days,  interest  $5; 
required  the  rate. 

Interest  on  $300  for  60  days  at  Q  %  =  $3.  At 
1  ^  =rz  .50.  It  is  obvious  that  the  rate  will  be  as 
great  as  the  number  of  times  1  ^  is  contained  in  the 
interest  given.     Hence,  $5.00 -f- 50  =  the  rate,  10  ^. 

Proof.— Interest  on  $300  for  60  days  at  10^  =$5. 

Answers:  $293.42,  $219.52,  10  %,  $173.30,  $126.32, 

$722.66. 


*The  learner  can   prove  his  work   by  computiag   interest   on   the 
principal   found. 


110  COMPLEX   PERCENTAGE. 

2.  The  principal  is  $396.15,  time  13  months  9  days, 
interest  $26.34,3 ;  required  the  rate. 

3.  What  is  the  rate  per  cent,  on  $144  for  5  days, 
when  the  interest  is  24  cents? 

4.  Eequired  the  rate  on  $250  for  60  days,  when  the 
interest  is  $3.50. 

5.  The  principal  is  $820,  time  30  days,  interest 
86.15  ;  what  is  the  rate? 

Art.  12.  To  find  the  Time,  when  the  principal^  rate 
per  cent  and  interest  are  given. 

Grace  being  allowed  only  on  notes  and  drafts, 
where  neither  is  named  it  is  not  reckoned. 

1.  The  principal  is  $1440,  rate  10  ^  per  annum, 
interest  $37.50  ;  required  the  time. 

Interest  on  $1440  for  1  day  at  10  %  =  40  cents. 

Since  there  are  as  many  days  as  the  interest  for  1 
day  is  contained  times  in  the  interest  given,  $37.50  -^ 
40  =:  93f ,  or  94  days. 

Proof. — Interest  on  $1440  for  94  days  at  10  %  per 
annum  =  $37.60.* 

2.  The  principal  is  $1674,  rate  2  ^  per  month,  in- 
terest $59.87;  required  the  time. 

3.  In  what  time  will  a  note  for  $600,  at  6  ^  per 
annum,  draw  $27.50  interest? 

4.  A  note  for  $375  drew  $21  interest  at  6  ^  per 
annum ;  how  long  did  it  require  to  do  it? 

5.  A  merchant  wishes  to  know  the  time  it  will  take 
a  balance  of  $917.50  to  make  $60.80,  with  interest  at 
10%. 

Answers:  8|-%,  12%,  6^  11  months  6  days,  54 
days,  108  days,  239  days,  94  days,  275  days.  9%. 

-'Interest  is  never  reckoned  on  the  fraction  of  a  day,  hence  the 
difference. 


COMPLEX     PERCENTAGE.  Ill 

Art.   13.      MISCELLANEOUS    EXEKCISES. 

1.  What  is  the  gain  per  cent,  on  goods  bought  at 
SI. 20  and  sold  at  $1.35? 

135  =  8elling  price. 
120=cost  price. 

15= actual  gain. 

j^nd*y^^^(5C.=gain  on  one  cent.  On  100  cents  there 
will  be  100  times  J/^j  or  i-520_o  =  i2_6_  or  12^. 

Note. — If  the  gain  is  an  even  part  of  the  first  cost  take  the 
same  part  of  100.  This  is  the  reverse  of  operation  1st  in  last 
Article. 

In  the  present  Ex.  15  is  J  of  120,  therefore  the  gain  per  cent,  is 
\  of  100  or  12^. 

FIRST  COST.  SELLING  PRICE.  GAIN  PER  CENT 

2.  $2.00  $3.00 

3.  1.25  1.50 

4.  0.75  1.00 


5.     0.10  0.12^ 


Total,     1281 


What  was  the  first  cost  of  the  goods   marked  a» 
follows? 


9.     3.75    @  25  %  loss. 

10.  0.87^  ''  12»  %     " 

11.  0.12^  "  50    %     " 


6.  115.87  @  12^  %  gain. 

7.  14.54  '^  3^  %     " 

8.  00.87  "  16|  %     " 

12.  A  bill  of  $1687. 75, had  been  reduced  10%, 
what  was  the  original  amount? 

13.  A  carpenter  puts  in  an  estimate  at  25  %  off 
the  bill  of  prices,  and  another  puts  one  at  10  %  off 
the  first;  how  much  per  cent,  off  the  bill  was  his 
discount  ? 

Answers :  12^  %,  50  %,  20  %,  33J  %,  25  %,  $102.97, 
$10.90,  $0.74,  $5,  $0.25,  $1,  1875.28,  32^  %. 


112  COMPLEX  PERCENTAGE. 

14.  Bought  a  bbl.  of  apples  for  $1.75,  and  sold  it  for 
$2.25  ;  what  did  I  gain  per  cent? 

15.  Sold  25  bbls.  potatoes  for  $39  ;  how  much  did  I 
gain  per  cent,  if  they  cost  me  $1.25  per  barrel? 

16.  Bought  150  bbls.  of  flour  @  $5.25,  paid  for 
drayage  $7.50,  and  porterage  $1 ;  at  what  per  barrel 
should  I  sell  it  to  gain  15  %. 

17.  Bought  15  horses  at  $125  each  and  sold  the  lot 
for  $3500 ;  what  was  my  gain  per  cent,  after  paying 
$25  for  their  feed  ? 

18.  Sold  a  safe  which  cost  me  $80  for  $75;  what 
•  was  my  loss  per  cent.  ? 

19.  Bought  a  bill  of  goods  for  $350,  paid  freight 
$15.20 ;  insurance,  $5 ;  drayage,  $3 ;  and  sold  them 
for  $425.  What  was  my  actual  gain,  and  what  my 
gain  per  cent.  ? 

20.  Bought  Henry  Ullhorn's  note  for  $750  at  a 
discount  of  15  ^  ;  what  did  1  pay  for  it? 

21.  Sold  Henry  Hazin's  note  of  $320  for  $300; 
what  was  the  rate  of  discount? 

22.  A  jeweler,  whose  business  capital  is  $10000, 
makes  100  %  on  his  goods,  and  takes  in  on  an  aver- 
age $20  a  day.  A  grocer,  whose  capital  is  $1000, 
profits  15  %,  and  takes  in  $35  a  day.  The  jeweler's- 
expenses  being  $1000,  and  the  grocer's  $300  per  year; 
what  does  each  one  gain  %  on  the  capital  invested  ? 

23.  A  pork  merchant  receives  a  quantity  of  pork  to 
be  sold  on  commission,  at  2J  ^;  or  he  may  have  the 
whole  on  his  own  account  at  7-^  cents  per  pound ; 
should  he  sell  on  commission,  or  buy,  supposing  he 
can  get  8|^  cents  a  pound? 

Answers:  28f  %,  $6.10,  84^  ^,  6i  ^,  14  %, 
$51.80,  $637.50,  6^  %,  $6500,  lllf  %,  buy. 


COMPLEX    PERCENTAGE.  113 

24.  A  huckster  commences  business  on  $50,  turns 
his  money  every  3  days,  making  2  cents  on  every  10, 
how  much  does  he  make  in  the  year,  provided  he 
spends  $15  a  month  for  rent,  and  puts  out  his  gains 
at  6  %  interest  at  the  end  of  every  month  ? 

25.  A  bookkeeper  who  receives  a  salary  of  $1500  a 
year,  and  loans  his  emplo3^er  $2000  at  10  ^,  is  offered 
a  fourth  of  the  profits  on  $8000  for  five  years,  for  his 
capital,  influence  and  services;  would  he  gain  or  lose 
by  accepting  the  offer,  the  profits  of  the  business  be- 
ing 20  %  per  annum  ? 

26.  The  assignee  of  an  insolvent  debtor  reports  to  the 
court  that  preferred  claims  (which  must  be  paid  in  full 
before  the  general  creditors  are  entitled  to  a  dividend) 
have  been  proven  to  the  amount  of  $386 ;  other  claims, 
$40630 ;  that  he  has  realized  from  collections  and  sales, 
$8650.  The  costs  of  court  to  date  are  $8.50;  the  fee 
for  assignee's  counsel,  $50 ;  assignee's  commission  on 
the  cash  reported,  d%  ;  auctioneer's  commission,  2% 
on  $3260  sales.  Give  the  assignee's  and  auctioneer's 
commissions,  the  per  cent  of  dividend  (without  a  frac- 
tion) that  can  be  paid  to  the  general  creditors,  and  the 
balance  of  cash  that  will  remain  in  the  hands  of  the 
assignee  after  paying  costs,  fees,  commissions,  preferred 
claims,  and  dividend  to  general  creditors. 

27.  The  final  report  of  the  assignee  in  the  above  case, 
shows  that  all  the  property  of  the  assignor  has  been  re- 
duced to  cash  ;  that  there  is  in  his  hands  $10680.  The 
unpaid  costs  are  $18.60 ;  assignee's  commission  on  the 
money  reported,  less  the  balance  on  hand  at  last  report, 
5%;  assignee's  attorney's  fee,  $200;  sundry  expenses  of 
the  trust,  $196.92.  What  is  the  assignee's  commission 
at  this  settlement,  and  how  much  will  the  general  credr 
itors  receive  on  the  dollar  ? 

Answers:    $65.20,  $432.50,  18  %,  $394.40,  $412.78, 
28  %,  $514.28,  Lose  $6500,  $1048.05. 


114  XXII.  TIME  TABLE 

FOB   COMPUTING   INTEREST   AND   AVERAGE. 

Number  of  days  from  \st  of  January  to  any  other  day  of  the  year.    In  lenp-years^ 
add  1  to  the  days  after  28th  of  February. 


o 

1 

1 

0 

2 

1 

3 

2 

4 

3 

5 

4 

6 

5 

7 

6 

8 

7 

9 

8 

10 

9 

11 

10 

12 

11 

13 

12 

14 

13 

15 

14 

16 

15 

17 

16 

18 

17 

19 

18 

20 

19 

21 

20 

22 

21 

23 

22 

24 

23 

25 

24 

26 

25 

27 

26 

28 

27 

29 

28 

30 

29 

31 

130 

~9(T 

91 

92 

93 

94 

95 

96 

97 

98 

99 

100 

101 

102 

103 

104 

105 

106 

107 

108 

109 

110 

111 

112 

113 

U4 

115 

116 

117 

118 

119 


31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 


59 

60 
61 
62 
63 
64 
65 
66 
67 
68 
69 
70 
71 
72 
73 
74 
75 
76 
77 
78 
79 
80 
81 
82 
83 
84 
85 
86 
87 
88 
89 


120 
121 
122 
123 
124 
125 
126 
127 
128 
129 
130 
131 
132 
133 
134 
135 
136 
137 
138 
139 
140 
141 
142 
143 
144 
145 
146 
147 
148 
149 
150 


151 
152 
153 
154 
155 
156 
157 
158 
159 
160 
161 
162 
163 
164 
165 
166 
167 
168 
169 
170 
171 
172 
173 
174 
175 
176 
177 
178 
179 
180 


111181 


2 

3 

4 

5 

6 

1 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 


2121243 


182 
183 
184 
185 
186 
187 
188 
189 
190 
191 
192 
193 
194 
195 
196 
197 
198 
199 
200 
201 
202 
203 
204 
205 
206 
207 
208 
209 
210 
211 


213 
214 
215 
216 
217 
218 
219 
220 
221 
222 
223 
224 
225 
226 
227 
228 
229 
230 
231 
232 
233 
234 
235 
236 
237 
238 
239 
240 
241 
242 


244i 
245 
246 
247' 
248 
249 
250 
251 
252 
253 
254 
255 
256 
257 
258 
259 
260 
261 
262 
263 
264 
265 
266 
267 
268j 
269 
270 
271 
272 


273 
274 
275 
276 
277 
278 
279 
280 
281 
282 
283 
284 
285 
286 
287 
288 
289 
290 
291 
292 
293 
294 
295 
296 
297 
298 
299 
3001 
301 
302 
303 


304 
305 
306 
307 
308 
309 
310 
311 
312 
313 
314 
315 
316 
317 
318 
319 
320 
321 
322 
323 
324 
325 
326 
327 
328 
329 
330 
331 
332 
333 


334 
335 
336 
337 
338 
339 
340 
341 
342 
343 
344 
345 
346 
347 
348 
349 
350 
351 
352 
353 
354 
355 
356 
357 
358 
359 
360 
361 
362 
363 
364 


TIME   TABLE  115 

FOR  COMPUTING  INTEREST  AND  AVERAGE. 

Nvmber  of  days  from  1st  of  Juty  to  any  other  day  of  the  year.     In  leap-years^  add  1  t* 
the  days  after  2Sth  of  February. 


p 

r 
(w 
p 

f 

c 

o 

1 

i 

u 

1 
1 

a* 

p 

> 

p 

a 

o 

p 

•-i 

1 

Z 
^ 

o 

^ 

? 

s 

p 

^f 

0 

31 

M 

92 

123 

153 

1 

184 

215 

243 

274 

304 

335 

1 

2 

1 

32 

63 

93 

124 

154 

2 

185 

216 

244 

275 

305 

336 

2 

3 

2 

33 

64 

94 

125 

155 

3 

186 

217 

245 

276 

306 

337 

3 

4 

3 

34 

65 

95 

126 

156 

4 

187 

218 

246 

277 

307 

338 

4 

5 

4 

35 

66 

96 

127 

157 

5 

188 

219 

247 

278 

308 

339 

5 

6 

5 

36 

67 

97 

128 

158 

6 

189 

220 

248 

279 

309 

340 

6 

7 

6 

37 

68 

98 

129 

159 

7 

190 

221 

249 

280 

310 

341 

7 

8 

7 

38 

69 

99 

130 

160 

8 

191 

222 

250 

281 

311 

342 

8 

9 

8 

39 

70 

100 

131 

161 

9 

192 

223 

251 

282 

312 

343 

9 

10 

9 

40 

71 

101 

132 

162 

10 

193 

224 

252 

283 

313 

344 

10 

11 

10 

41 

72 

102 

133 

163 

11 

194 

225 

253 

284 

314 

345 

11 

12 

11 

42 

73 

103 

134 

164 

12 

195 

226 

254 

285 

315 

346 

12 

13 

12 

43 

74 

104 

135 

165 

13 

196 

227 

255 

286 

316 

347 

13 

14 

13 

44 

75 

105 

136 

166 

14 

197 

228 

256 

287 

317 

o48 

14 

15 

14 

45 

76 

106 

137 

167 

15 

198 

229 

257 

288 

318 

349 

15 

16 

15 

46 

77 

107 

138 

168 

16 

199 

230 

258 

289 

319 

350 

16 

17 

16 

47 

78 

108 

139 

169 

17 

200 

231 

259 

290 

320 

351 

17 

18 

17 

48 

79 

109 

140 

170 

18 

201 

232 

260 

291 

321 

352 

18 

19 

18 

49 

80 

110 

141 

171 

19 

202 

233 

261 

292 

322 

353 

19 

20 

19 

50 

81 

111 

142 

172 

20 

203 

234 

262 

293 

323 

354 

20 

21 

20 

51 

82 

112 

143 

173 

21 

204 

235 

263 

294 

324 

355 

21 

22 

21 

52 

83 

113 

144 

174 

22 

205 

236 

264 

295 

325 

356 

22 

23 

22 

53 

84 

114 

145 

175 

23 

206 

237 

265 

296 

326 

357 

23 

24 

23 

54 

85 

115 

146 

176 

24 

207 

238 

266 

297 

327 

358 

24 

25 

24 

55 

86 

116 

147 

177 

25 

208 

239 

267 

298 

328 

359 

25 

26 

25 

56 

87 

117 

148 

178 

26 

209 

240 

268 

299 

329 

360 

26 

27 

26 

57 

88 

118 

149 

179 

27 

210 

241 

269 

300 

330 

361 

27 

28 

27 

58 

89 

119 

150 

180 

28 

211 

242 

270 

301 

331 

362 

28 

29 

28 

59 

90 

120 

151 

181 

29 

212 

271 

302 

332 

363 

29 

30 

29 

60 

91 

121 

152 

182 

30 

213 

272 

303 

333 

364 

30 

31 

30j 

61 

122 

183 

31 

214 

273 

334 

31 

116  PAST-TIME  TABLE 

FOR   COMPUTING  INTEREST  AND   AVERAGE. 

Number  of  days  frovi  Jannary  1st  to  any  day  of  the  previous  year, 
pmrs  add  one  day  before  February  28th. 


In  leap' 


1 

o 

i 

1 

1 
1 

> 

g 

e-i 

c 

0 

1 
1 

o 

1 

o 

B 

a- 

1 

o 

1 

p- 

"T 

365 

334 

306 

275 

245 

214 

~1 

184 

153 

122 

92 

61 

31 

1 

2 

364 

333 

305 

274 

244 

213 

2 

183 

152 

121 

91 

60 

30 

2 

3 

363 

332 

304 

273 

243 

212 

3 

182 

151 

120 

90 

59 

29 

3 

4 

362 

331 

303 

272 

242 

211 

4 

181 

150 

119 

89 

58 

28 

4 

5 

361 

330 

302 

271 

241 

210 

5 

180 

149 

118 

88 

57 

27 

5 

6 

360 

329 

301 

270 

240 

209 

6 

179 

148 

117 

87 

56 

26 

6 

7 

359 

328 

300 

269 

239 

208 

7 

178 

147 

116 

86 

55 

25 

7 

8 

358 

327 

299 

268 

238 

207 

8 

177 

146 

115 

85 

54 

24 

8 

9 

357 

326 

298 

267 

237 

206 

9 

176 

145 

114 

84 

53 

23 

9 

10 

356 

325 

297 

266 

236 

205 

10 

175 

144 

113 

83 

52 

22 

10 

11 

355 

324 

296 

265 

235 

204 

11 

174 

143 

112 

82 

51 

21 

11 

12 

354 

323 

295 

264 

234 

203 

12 

173 

142 

111 

81 

50 

20 

12 

13 

353 

322 

294 

263 

233 

202 

13 

172 

141 

110 

80 

49 

19 

13 

14 

352 

321 

293 

262 

232 

201 

14 

171 

140 

109 

79 

48 

18 

14 

15 

351 

320 

292 

261 

231 

200 

15 

170 

139 

108 

78 

47 

17 

15 

16 

350 

319 

291 

260 

230 

199 

16 

169 

138 

107 

77 

46 

16 

16 

17 

349 

318 

290 

259 

229 

198 

17 

168 

137 

106 

76 

45 

15 

17 

18 

348 

317 

289 

258 

228 

197 

18 

167 

136 

105 

75 

44 

14 

18 

19 

347 

316 

288 

257 

227 

196 

19 

166 

135 

104 

74 

43 

13 

19 

20 

346 

315 

287 

256 

226 

195 

20 

165 

134 

103 

73 

42 

12 

20 

21 

345 

314 

286 

255 

225 

194 

21 

164 

133 

102 

72 

41 

11 

21 

22 

344 

313 

285 

254 

224 

193 

22 

163 

132 

101 

71 

40 

10 

22 

23 

343 

312 

284 

253 

223 

192 

23 

162 

131 

100 

70 

39 

9 

23 

24 

342 

311 

283 

252 

222 

191 

24 

161 

130 

99 

69 

38 

8 

24 

25 

341 

310 

282 

251 

221 

190 

25 

160 

129 

98 

68 

37 

7 

25 

26 

340 

309 

281 

250 

220 

189 

26 

159 

128 

97 

67 

36 

6 

26 

27 

339 

308 

280 

249 

219 

188 

27 

158 

127 

96 

66 

35 

5 

27 

28 

338 

307 

279 

248 

218 

187 

28 

157 

126 

95 

65 

34 

4 

28 

29 

337 

278 

247 

217 

186 

29 

156 

125 

94 

64 

33 

3 

29 

30 

336 

277 

246 

216 

185 

30 

155 

124 

93 

63 

32 

2 

30 

31 

335 

276 

1 
1 

215 

31 

154 

123 

62 

1 

31 

PAST-TIME  TABLE 

FOR   COMPUTING  INTEREST   AND   AVERAGE. 
Number  of  days  from  July  \st  to  any  day  in  the  past  year. 


117 


o 

e^ 

> 

w 

o 

!2; 

Q 

o 

tH 

>^ 

K 

>■ 

ss: 

t_ 

o 

S 

s 

a 
at 

a 

(6 
1 

B 
1 

o 

o 

B 

1 

o 

a 

a 
1 

1 

p. 

I 

^ 

c 
a 

J 

1 

365 

334 

303 

273 

242 

212 

1 

181 

150 

122 

91 

61 

30 

T 

2 

364 

333 

302 

272 

241 

211 

2 

180 

149 

121 

90 

60 

29 

2 

3 

363 

332 

301 

271 

240 

210 

3 

179 

148 

120 

89 

59 

28 

3 

4 

362 

331 

300 

270 

239 

209 

4 

178 

147 

119 

88 

58 

27 

4 

5 

361 

330 

299 

269 

238 

208 

5 

177 

146 

118 

87 

57 

26 

5 

6 

360 

329 

298 

268 

237 

207 

6 

176 

145 

117 

86 

56 

25 

6 

7 

359 

328 

297 

267 

236 

206 

7 

175 

144 

116 

85 

55 

24 

7 

8 

358 

327 

296 

266 

235 

205 

8 

174 

143 

115 

84 

54 

23 

8 

9 

357 

326 

295 

265 

234 

204 

9 

173 

142 

114 

83 

53 

22 

9 

10 

356 

325 

294 

264 

233 

203 

10 

172 

141 

113 

82 

52 

21 

10 

11 

355 

324 

293 

263 

232 

202 

11 

171 

140 

112 

81 

51 

20 

11 

12 

354 

323 

292 

262 

231 

201 

12 

170 

139 

111 

80 

50 

19 

12 

13 

353 

322 

291 

261 

230 

200 

13 

169 

138 

110 

79 

49 

18 

13 

14 

352 

321 

290 

260 

229 

199 

14 

168 

137 

109 

78 

48 

17 

14 

15 

351 

320 

289 

259 

228 

198 

15 

167 

136 

108 

77 

47 

16 

15 

16 

350 

319 

288 

258 

227 

197 

16 

166 

135 

107 

76 

46 

15 

16 

17 

349 

318 

287 

257 

226 

196 

17 

165 

134 

106 

75 

45 

14 

17 

18 

348 

317 

286 

256 

225 

195 

18 

164 

133 

105 

74 

44 

13 

18 

19 

347 

316 

285 

255 

224 

194 

19 

163 

132 

104 

73 

43 

12 

19 

20 

346 

315 

284 

254 

223 

193 

20 

162 

131 

103 

72 

42 

11 

20 

21 

345 

314 

283 

253 

222 

192 

21 

161 

130 

102 

71 

41 

10 

21 

22 

344 

313 

282 

252 

221 

191 

22 

160 

129 

101 

70 

40 

9 

22 

23 

343 

312 

281 

251 

220 

190 

23 

159 

128 

100 

69 

39 

8 

23 

24 

342 

311 

280 

250 

219 

189 

24 

158 

127 

99 

68 

38 

7 

24 

25 

341 

310 

279 

249 

218 

188 

25 

157 

126 

98 

67 

37 

6 

25 

26 

340 

309 

278 

248 

217 

187 

26 

156 

125 

97 

66 

36 

5 

26 

27 

339 

308 

277 

247 

216 

186 

27 

155 

124 

96 

65 

35 

4 

27 

28 

338 

307 

276 

246 

215 

185 

28 

154 

123 

95 

64 

34 

3 

28 

29 

337 

306 

275 

245 

214 

184 

29 

153 

94 

63 

33 

2 

29 

30 

336 

305 

274 

244 

213 

183 

30 

152 

93 

62 

32 

1 

30 

31 

335 

304 

243 

182 

31 

151 

92 

31 

31 

118  AVERAGE. 


XXIII.  AVERAGE. 


Art.  1.  When  several  payments  have  to  be  made 
at  one  time,  or  when  one  bill  has  to  be  paid  v^^ith 
several  notes  of  different  lengths  of  time,  an  average 
has  to  be  sought,  the  process  of  finding  which  is 
called  Average,  or  Equation  of  Payments. 

1.  A  merchant  sells  a  bill  of  goods  amounting  to 
«4000,  to  be  paid  as  follows :  $400  in  30  days,  $600  in 
(lO  days,  $1000  in  90  days,  and  the  balance  in  4  mos., 
or  120  days,  what  would  be  a  mean  or  average  time 
of  payment  for  the  whole? 

A  credit  of  $400  for   30  ds.,  is  the  same  as  a  credit  on  $1  for  12000  ds- 
»  600  "      60  "         "         "         "         "         1  "    36000  " 

"         1000  ''     90  "         "         «         "         "         1  "   90000  " 
"         2000  "   120  "        "        "        "        «        1  "240000  " 


4000  378000 

On  one  dollar  there  is  a  credit  for  378000  days. 
On  $4000,  there  is  a  credit  for  ^U^^=^H  ^^J^- 
That  is,  the  $4000  might  be  paid  in  94^  days^  or  on 

the   95th  day,   without  either  party  sustaining  loss 

by  interest. 

2.  A  merchant  sells  goods  to  the  amount  of  $1700, 
$500  payable  in  60  days,  $300  payable  in  90  days, 
and  $900  payable  in  30  days,  what  is  the  average 
time  of  payment  of  the  whole? 

3.  Sold  a  bill  of  goods,  amounting  to  $700,  J  of 
which  is  payable  in  90  days,  J  in  4  mos.,  and  ^  in  6 
mos.;  required  the  average  time  of  payment. 

Answers:  49  days,  143  days. 


AVERAGE.  119 

Art.  2.     To  find  the  average  date  of  purchase. 

1.  Purchased  goods  as  follows,  what  was  the  average 
date  of  purchase? 

Dec.  31,  a  bill  of  $300,  Jan.  3,  a  bill  of  $100,  Jan.  9, 
a  bill  of  $200,  Jan.  18,  a  bill  of  $800,  Jan.  23,  a  bill* 
of  $500.  '  Ans.  Jan.  15th. 


Remark. — If  the  amounts  abo"ve  were  equal,  and  the  intervals 
also  equal,  the  average  date  of  purchase  would  be  on  Jan.  9th: 
because  it  is  midway  between  the  first  and  last  dates. 

ExPL. — The  first  was  due  at  the  time  of  purchase;  the  second, 
3  days  after;  the  third,  9  days  after,  etc. 

300  X   0  = 
100 X   3=     300 
200  X    9=  1800 
800X18  =  14400 
500X23=11500 


1900  )28O00 

14i|,  or  15  days  after  Dec.  31, 
the  date  of  first  purchase, 
which  brings  the  time  up  to 
Jan.  15  th. 

If  these  debts  had  been  contracted  on  a  credit  of 
three  months,  a  note  dated  Jan.  15  would  be  given 
to  settle  the  bill. 

2.  What  is  the  average  date  of  purchase  of  the  fol- 
lowing ? 

Jan.  1,  Mdse.,  $360,  Feb.  6,  Mdse.,  $325,  March  8, 
Mdse.,  $180,  April  3,  Mdse.,  $65,  May  13,  Mdse.,  $275, 
June  8,  Mdse.,  $70. 

Am,  Jan.  15,  March  3,  Feb.  26. 


120 


AVERAGE. 


§1000.00 

3500.00 

9734.00 

976.50 

1037.00 


3.  The  following  goods  were  sold  on  a  credit  of 
hO  days : 

JSTew  York,  Apr.  3,  1876. 
Mr.  James  Callen,  Bought  of  Eobt.  Boggs. 

Jan.     1,  Invoice  of  Coffee,  ., 
^'         6,        '*         *^   Sugar,'  . 
Mar,    9,        <'         ''  Sunds.,  . 

Apr.     3,        "         '' 

1T6247750 
Kequired  the  average  date  of  purchase,  or  date  of 
note. 

4.  Philadelphia,  Dec.  3,  1859. 
Mr.  Henry  Higgins, 

Bought  of  James  Kiel, 
Sept.    3,  Invoice  of  Calicoes,        ....     §3150.00 

^'      19,         "         ^'  Muslins, 1174.00 

'^      20,         "         "  Silks, 3500.00 

Oct.    19,         ''         "  Sundries,       ....       1743.00 

$9567700 
Required  the  date  of  maturity  of  a  3  months'  note, 
grace  included. 

Find  the  equated  time  of  payment  for  the  follow- 
ing: 


5. 


6. 


Apr.    3, 

$167.25* 

May     7, 

$674.40 

-       9, 

374.00 

Jun.     7, 

168.37 

^'     19, 

176.00 

"      10, 

370.20 

-     20, 

371.00 

'^      15, 

167.00 

«     25, 

•    197.87 

"      19, 

679.60 

''     30, 

300.00 

July  23, 

679.45 

May   9, 

150.57 

•    Aug.  18, 

993.18 

^'     23, 

720.18 

"      19, 

875.57 

Answers : 

Feb.  22,  April  30, 

July  11,  Dec 

.  23. 

^When  the  cents  are  under  50,  reject  tliem,  otherwise  add  a 
dollar  to  the  dollars. 


AVERAGE.  121 

Art.  3.  Whe7i  goods  are  pvrchased  at  different  dates 
and  on  different  lengths  of  credit. 

1.  Purchased  the  following  bills  of  merchandise; 
required  the  average  date  of  maturity,  or  the  equated 
time  of  payment  for  all : 

Apr.  3,  a  bill  of  $250  on  3  months'  credit. 
a        g         a        u       jgy    a     g         u  u 

May  1,      «      "      250  ''    4       "  " 

Jun.  9,      "      "      320  ''2       "  .  " 

If  we  substitute  the  date  of  maturity  of  each  of 
these  bills  for  the  date  of  purchase^  and  arrange  them 
in  the  order  of  time,  we  shall  have  a  problem  in  all 
respects  similar  to  those  under  last  Art. 

The  first  bill  falls  due  July  3d,*  the  second  Oct. 
9th,  the  third  Sept.  7th,  the  fourth  Aug.  9th.  Ar- 
ranged  in  the  order  of  time,  they  appear  thus  : 

July   3,  $250 

Aug.  9,  320X37  =  11840 

Sept.  7,  250X66  =  16500 

Oct.    9,  157X98=15386 

977  43726(44T[|4  or  45  days. 

3908 


4646 

3908 


9TT 


Hence  the  date  of  maturity  is  45  days  after  July 
3d,  or  on  Aug.  17th  ;  from  which  time  till  the  day 
of  settlement,  interest  is  due  on  the  whole  amount. 


*  Days  of  grace  are  not  allowed  on  invoices. 


122  AVERAGE. 


When  some  of  the  purchases  are  at  cash  price. 

2.  What  is  the  equated  time  of  payment  for  the  fol- 
lowing: Jan.  1,  8600  on  3  mos.,  Feb.  3,  1670  at  cash 
price,  Mar.  3,  $950  on  6  mos.,  May  3,  S550  for  cash? 

The  first  payment  falls  due  on  April  1st,  the  second, 
being  for  cash,  was  due  at  the  time  of  purchase,  Feb. 
3d,  the  third  Sept.  3d,  and  the  last  May  3d.  Ar- 
ranging the  dates  and  amounts  in  the  order  of  time, 
«8  before,  we  have 


Feb.  3, 

$670 X  00 

Apr.  1, 

600 X  57 

34200 

May  3, 

550 X  89 

48950 

Sept.  3, 

950X212 

201400 

2770  284550(103  days  nearly,  or 

\lth  May, 

Note. — The  amount  due  on  Sept.  Sd,  is  $2770  plus  the  interest  on 
that  amount,  from  May  17th. 

From  May  17th.  to  Sept.  .3d.  =  109  days.  Interest  on  $2770  for 
109  days  =  $50.82,  which,  added  to  $2770  =  |2820.:32. 

Find  the  equated  time  of  payment  of  the  following; 

3.  July  1,  $675  on  3  mos. ;  13th,  $619.54  on  2  mos. ; 
19th,  $147.67  at  cash  rates  ;  23d,-  $678.44  on  5  mos. 

4.  Sept.  3d,  $937.15  on  30  days' credit;  9th,  $897.78 
on  90  days'  credit;  17th,  $619.18  at  cash  prices; 
Oct.  3d,  $777  on  60  days. 

Eequired  the  amount  due  on  each  of  the  following 
on  July  1st. 

5.  Jin.  9th,  $678.44  @  60  days;  20th,  $419.88  at 
cash  price;  29th,  $789.14  at  3  mos. 

Answers:  October  17,  Nov.  2,  May  17,  $1919.85, 
$4708.49. 


AVERAGE.  123 


6.  April  9th,  $1678  on  3  mos.  ;  June  18,  $1000  at 
cash  prices;  21st,  $879.55  on  60  days;  23d,  $371.19 
cash  ;  20th,  $785.25  cash.  A?is.  $4708.49. 

Art.  4.  APPLICATION  TO  ACCOUNT  SALES. 

An  Account  Sales  is  a  detailed  statement  of  goods 
sold  by  a  commission  merchant,  on  account  of  the 
party  who  sent  them. 

The  person  or  party  who  sends  goods  to  another 
to  be  sold  for  himself,  is  called  tho  consignory  the  per- 
son to  whom  they  are  sent,  the  consignee,  and  the 
goods  sent,  the  consignment. 

COMMISSION  HOUSE  OF  STRAIGHT,  DEMING  &  CO. 

Shipment  18.  No.  7828. 

Sales  for  account  of  Messrs.  Gaff  &  Baldwin. 

By  sundries, 
Jun.    4,  T.  B.  Colgan  &  Co.  @  60  days,  8  hhds  Sugar. 

1095  1020 

1100  1120 

1080  1240     8965 

1200  1110       896     8069       @7/g  $600.13 

Jun.  6,  G.  Newton  &  Co.,®  60  days,  10  hhds.  Sugar. 

1080  1040 

•  1090  1340 

1120  1020 

1240  1100     11440 

1200  1210       1144  10296    @  6|     $707.85 


124  AVERAGE. 


Sales  for  account  of  Messrs.   Gaff  &  Baldwin — continued. 
Jun.  10,  B.  Yilgers  k  Co.,  @  60  days,20  hhds.  Sugar. 

1060  1240 

1210  1110 

1180  1005 

1055  1285 

1240  1100 

1185  1210 

1300  1325 

•   1010  1140 

1120  1205  23185 

1205  1000   2318 


20867  ©6^5  1343.31 
$2651.29 


Charges, 

Jun.  1,  Fd  cash  st'mrLandis  for  fr't,$  87.18 
''      *'    Drayage,  9.50 

"     14,  Insurance,  4.63 

**      ''    Storage,  9.50 

**      "    Commission  and  guarantee,    132.56       243.37 


Netproc'ds  due  by  equa^n^  Aug,  13,         $2407.92 
E.  O.  E. 
Cincinnati,  June  14th,  1876. 

Straight,  Deming  &  Co., 
per  F.  Jelke. 

Recapitulation  of  answers  of  Art.  1-4  inclusive  :  Jan. 
15,  Feb.  22.  April  30,  May  17,  July  11,  Aug.  7,  Aug. 
17,  Oct.  17,  Nov.  2,  Dec.  23.  45  days,  49  days,  95 
days,  143  days,  $1919.85,  $4708.49. 


AVERAGE.  125 

Art.  5.  When  payments  are  made  before  a  note 
or  bill  is  due,  to  find  how  long  after  maturity  it 
should  run,  to  balance  the  interest  on  the  advanced 
payments. 

1.  A  merchant  holds  a  note  of  $500  at  6  months. 
Three  months  before  it  is  due,  he  receives  $100,  and 
one  month  before  it  is  due,  he  receives  $300,  how 
long  should  he  allow  the  balance  to  run,  to  equal 
the  interest  on  the  advance? 

The  int.  on  $100  for  3  mo8.=int.  on  $1  for  300  mos. 
"         "     300    u    1     u     _  a      u      I    u  300     " 

600  mos. 

Hence,  the  interest  on  the  advanced  payments  is 
equal  to  the  interest  of  $1  for  6^0  months  ;  that  is, 
a  balance  of  $1  should  have  run  600  months,  but  the 
balance  due  on  the  note  is  $100;  therefore,  it  should 
run  fg§  months  =6  mos. 

Proof. — The  int.  of  the  $100  that  is  to  run  6  mos. =$3. 

The  int.  of  $100  (the  first  pay't,)  for  the  3  mos.=$1.50 
"         "    300  (the  sec.  pay' t,)  "      "1     ^^    =  1.50 

Total  interest  on  advance  =$3.00 

2.  A  note  of  $600  given  on  Jan.  3,  1876,  payable  in 
6  months.  4  months  before  it  was  due,  $100  was 
paid  on  it,  and  3  months  before  it  was  due,  $200  was 
paid;  how  long  in  equity  should  the  balance  run? 

3.  A  merchant  owes  $700  due  8  months  from  the 
time  he  contracted  the  bill ;  5  months  afterward,  he 
pays  $200,  and  2  months  after  that,  $300,  how  long 
should  the  balance  remain  unpaid? 

Answers :  3  mo.  10  days,  4  mo.  15  days,  4  mo.  12 
days. 


126 


AVERAGE. 


4.  If  I  borrow  $600  from  A  at  one  time,  and  $500  at 
another,  each  for  4  months,  how  long  should  I  lend 
him  $1000  to  return  the  favor? 

Ans.  4  months  12  days. 

APPLICATION  TO  ACCOUNTS  CURRENT. 

Art.  6.  In  applying  equation  to  accounts  current, 
both  the  debit  and  credit  sides  of  an  account  have  to 
be  averaged,  for  which  reason,  the  operation  is  usually 
called  Compound  Equation. 


Dr. 


J.  Doe  in  acct  with  R.  Rob. 


Cr. 


$ 

c. 

* 

c. 

1876. 

1876. 

July 

3 

To  Mdse., 

1000 

00 

Aug. 

1 

By  Cash, 

500 

00 

(( 

7 

il          u 

500 

00 

u 

12 

u     u 

500 

00 

Aug. 

18 

U           (( 

250 

00 

Eequired  the  amount  due  on  Aug.  18th,  supposing 
the  sales  to  have  been  made  at  cash  rates. 

We  first  find  the  average  of  the  debit  side,  because 
it  contains  the  earlier. date.  The  credit  side  is  aver- 
aged from  the  same  date. 

July    B,  1000 X    0 

7,     500  X   4=  2000 
Aug.  18,     250X46  =  11500 

1750  )135O0(7j||,    or   8    days   from 

1225  July  3. 

"T25 


Aug.     1,  500X29  =  14500 
«      12,  500X40=20000 


1000 


)34500,  or  35  days  from  July  3. 


AVERAGE. 


127 


The  debit  side  averages  July  11th,  that  being  the 
mean  time  of  purchase.  The  credit  side  shows  that 
the  payments  average  Aug.  7th ;  hence  the  whole 
debt  was  due  from  July  11th  to  Aug.  7th  (27  days), 
for  which,  interest  should  be  charged;  and  the  bal- 
ance, $750,  was  due  from  Aug.  7th  to  Aug.  18th,  for 
which  interest  should  also  be  charged. 

Interest  on  $1750  for  27  days  =$7.88 
'*         «        750    "    11      "  1.38 


$9.26 

Which  added  to  the  debit  side  of  Doe's  account^ 
gives  a  balance  against  him  of  $759.26. 

Art.  7.  APPLIED    TO   STATEMENTS. 

When  settlement  is  made  by  note,  an  average  date  of 
payment  is  found  by  dividing  the  difference  of  the  prod- 
ucts by  the  balance  of  debt,  and  counting  backward  or 
forward  from  the  assumed  date. 

34500—13500=21000  days,  which  divided  by  750 
gives  28  days,  to  be  counted  backward  from  July  3d, 
giving  June  bth. 

Explanation. — In  the  preceding  pperation  we  assumed  that 
the  whole  debt  was  due  on  July  3d,  making  the  sales  subject  to 
a  discount  equal  to  that  on  $1  for  13500  days  (favor  of  buyer), 
and  the  payments  to  a  discount  of  34500  days  (against  buyer), 
showing  a  balance  against  him  of  21000  days  on  $1,  or  28  days 
on  his  debt  of  $750. 

To  be  against  him  we  must  count  backward,  for  to  count  for- 
ward would  give  him  longer  time  to  pay  the  note. 

Proof.— Interest  on  $750  from  June  5th  to  Aug.  18th  (74 
days),  $9.25. 

The  difference  of  1  cent  is  caused  by  the  fraction  of  a  day 
reckoned  as  a  full  day,  on  preceding  page. 


128 


AVERAGE. 


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tr    •       :j 

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£    i  £  = 

oil!: 

SJ58 

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•-2 

Is 

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1 

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5 

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r-""^        --■*        -•»"        --        -        -^        x" 

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t^  -;       ^             s-             ;-.                       >>                      r^ 

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AVERAGE. 


129 


Dr. 

A.  Mills. 

Ck. 

1876. 

$ 

c. 

1870. 

$ 

c. 

Feb. 

18 

To  Mdse, 

600 

00 

Apr. 

3 

By  Cash, 

600 

00 

(( 

28 

((         (( 

700 

00 

a 

12 

((  '     11 

400 

00 

Mar. 

17 

li              u 

800 

00 

Eequired   the    amount    due    July    Ist;    also,    the 
equated  time  at  which  a  note  would  have  been  dated. 

Art.  8.  APPLICATION  TO  STORAGE. 

Storage  is  usually  charged  for  by  the  box,  barrel, 
etc.,  for  the  month. 


Eeceived  July 
Eeceived     " 

1, 
4, 

bbls.         days,  products, 
200         X3=  600 
300  500X2=1000 

Delivered    " 
Balance       ^' 
Eeceived     " 
Eeceived     '^ 

6, 

10, 
16, 

100 
400         X4=1600 
300  700X6=4200 
200  900X2=1800 

Delivered    ' ' 
Balance       '< 
Eeceived     ^' 

18, 
21, 

600 
300         X3=  900 
500 

Balance  on  hand 

800 

30)10100 

336f 
The  products  divided  by  the  number  of  days  in  a 
month,   give    the  number  of  barrels  chargeable  for 
a  month. 


Answers :  $1229.20,  337  days. 
9 


180  AVERAGE. 


Art.  9.      GENERAL  EXERCISES. 

1.  A  bill  of  $1000  is  to  be  paid  in  five  equal  install- 
ments, at  3,  4,  5,  6,  and  7  months,  what  time  should 
be  allowed,  if  the  individual  will  pay  it  all  at  once? 

2.  The  following  bills  of  goods  have  been  pur- 
chased at  different  periods ;  required  the  average 
time  of  payment,  allowing  30  days  credit  on  each  : 

,  $1347  on  Jan.  1,  $167  on  Feb.  3,  $1794  on  Feb.  8, 
«6783  on  Feb.  10,  $1076  on  Feb.  19,  $319,  on  Mar.  6, 
and  $1674  on  April  9. 

3.  What  should  be  the  date  of  a  60  day  note  for 
the  following  bills:  $168  purchased  on  April  6, 
$3196  Apr.  9,  $1668  May  3,  $6847  June  1? 

4.  When  would  a  90  day  note  fall  due,  given  for 
the  following  bills:  $673  on  June  3,  $710  on  July  6, 
$415  on  July  9,  $678  on  Aug.  3? 

5.  What  is  the  storage  of  the  following  account, 
closed  Aug.  4,  at  5  cents  per  bbl.  per  month? 

Apr.    3,  received  167  July  1,  delivered  200 

"        9,         ''         145  "     3,         "           150 

"     17,         "        450  "    7,         "          190 

May  18,         "         198 

6.  At  5  cents  per  barrel,  what  will  the  storage  of 
the  following  amount  to  on  Oct.  24? 

July     9,  received  167  Oct.  1  delivered  125 

Aug.    5,         ''        378              "  3  "  500 

^*        9,         "         780               "  19  "  450 

*•      31  '•         178 

(Answers  on  next  page.) 


AVERAGE. 


131 


7.  How  much  was  due  on  the  following  account 
on  Sept.  13,  1859;  bills  sold  on  60  days  time:^ 

Dec.  8,  1858,  $1676,  Jan.  9,  1859,  $1675,  Feb.  14, 
paid  $500,  Apr.  16,  paid  $1000. 

No  credit  being  allowed  on  the  following  bills, 
required  the  balance  due  at  date  of  last  purchase 
jv  paj'nient. 

Debit  side.  Credit  side. 

8.  Apr.    9,  $600  May  13,  $700 

^'       9,     500  "     15,     100 

"     15,     700 

Balance  due  May  15, 


9 


June    3,  $365 
"     19,     784 
July  18,     594 

Balance  due  July  18, 
1858. 
10.  Dec.    7,  $1874 
1859. 

Jan     3,     1678 
''     21,       712.53 


June  20,  $300 
*'  29,  500 
"     30,    100 

1859. 
Jan.    7,  $1000 

April  14,       900 


What  will  be  average  date  of  maturity  of  the 
ollowing,  and  what  will  be  the  balance  due  Mar.  7? 

11.  Jan.    1,  $1673  on  3  mos.  credit,    Mar.  4,  $1000 
"       9,       740  on  2     "  "  *'     7,       500 

"     29,       500  on  4     ^'  " 

Answers:  March  16th,  May  13th,  Oct.  6th,  5  mo., 
$141.80,  $168.96,  $846.23,  $1009.93,  $1921.65, $2426.41, 
May  8th,  $1398.40. 

*  Some  reckon  every  fraction  of  a  day  as  a  whole  day;  others 
only  fractions  that  are  over  J^. 


132  METHODS    OF    AVERAGING. 


METHODS  OF  AVERAGING. 

There  are  two  methods  of  averaging  known  to  ac- 
countants, viz.:  the  Interest  Method  and  the  Product 
Method.  By  the  first  interest  is  fully  computed  upon 
every  item  up  to  the  day  of  settlement.  By  the  Prod- 
uct Method  the  time  is  usually  reckoned  from  the 
date  of  the  first  item,  and  multiplied  into  the  va- 
rious amounts.     From  this  arises  discount. 

A  new  method,  now  introduced  for  the  first  time, 
is  -a  modification  of  the  latter,  which  we  shall  call 
the  Interest- Product  Method.  By  it  the  time  is  reck- 
oned from  the  last  item  of  the  account,  or  the  day 
of  settlement,  which  results  in  giving  the  interest 
direct  without  further  calculation,  except  to  divide 
by  6000. 

The  Time  Tables  on  pages -114  and  115  are  used 
for  the  Discount-Prodtict  Method,  and  those  on 
pages  116  and  117  for  the  Interest-Product  Method. 

Pr.     •        12.     C.  A.  Walworth.  Or. 

1876.  1876. 

Jan.  1,  To  Mdse,        $300.00        Feb.    9,  By  Mdse,        $200.00 

Mar.  3,    "       ''               500.00                 23,    ''  "              100.00 

300X  0  200X39^7800 

500=61=30500  Dr.  products.  100X53=5300 

13100  Cr.  products.  

800 300          13100 

300     500)17400 


Bed.  due  500  34f  or  35  days  from  Jan.  1  or  Feb.  5. 

Explanation. — Assuming  both  purchases  and  sales  to  be 
due  on  January  1,  W.  would  be  entitled  to  a  discount  on  his 
purchases  equal  to  that  on  $1  for  30500  days,  and  I  would  be 
entitled  to  a  discount  on  my  purchases  equal  to  that  on  $1  for 
13100  days,  making  a  difference  of  17400  days  in  W.'s  favor  on 
the  balance,  $500.  The  discount  on  $1  for  17400  days  is 
equal  to  the  discount  on  $500  for  17400  days--500=34i  days 
from  January  1,  to  be  reckoned  forward,  which  will  give  Feb- 
ruary 5.  Had  the  discount  been  against  him,  it  would  have' 
■ehown   that  the  balance  was  past  due  on  January  1,  which 


AVERAGE.  135 


would  indicate  that  the  time  would  be  counted  backward  frona 
that  date.  Should  the  balance,  instead  of  the  equated  time, 
be  required,  the  difference  between  tlie  Dr.  and  Cr.  products 
may  be  divided  by  6000  to  find  the  discount  on  the  balance  ta 
the  day  of  settlement  at  the  rate  of  6  per  cent  per  annum.  Any 
amount  being  100  per  cent  of  itself,  when  multi{)lied  l)y  a  num- 
ber of  days,  will  be  100  per  cent  per  day.  Six  per  cent  per  an- 
num being  the  l-6000th  part  of  100  per  cent  per  day  ;  In-nce  by 
dividing  the  difference  as  aforesaid,  we  have  the  discc.unt  on 
the  balance  up  to  the  assumed  day  of  settlement  at  tiie  rate 
named.  17400  :  6000  =  $2.90  discount.  This  process  we  shall 
name  the  Discount- Product  Method. 


Dr.                      13.-       J.  C. 

HiNTZ. 

Or. 

1869. 

1869. 

July  3,  To  Mdse.      $1000.00 

Aug.    1,  By  Cash, 

Sooo.oa 

7,    "      "              500.00 

u      13^     u 

500.00 

Au.l8,    "      "              250.00 

Assuming  the  day  of.  settlement  to  be  July  1,  we 
have. 

lOOOX  2=  2000  Days.  500*X31  =15500  Days. 

500X  6=  3000       "  500X43=21500      " 


250X48=12000 

1000  37000  Cr.  products. 

1750  17000  "  17000  Dr.  products. 

1000  

20000  Difference  of  do. 

$750  Balance  due. 

750)20000(26  ds.      Explanation. — The  sum  of  the  credit  prod- 
150  ucts  being  greater  than  that  of  the  debit  prod- 

~5()0  ucts,  shows  that  the  discount  is  in  my  favor. 

450  Hence,  in  order  to  sett'e  on  tlie  assumed  date, 

-Tq  his  payments  would   be  at  a  discount- of  27 

days  for  the  balance;  but  as  it  would  be  im- 
possible to  settle  on  a  past  date,  we  will  have  to  charge  him  in- 
terest from  27  days  prior  to  July  1  (June  4)  to  the  real  day  of 
settlement,  whatever  that  may  be,  or  else  take  his  note,  dated 
June  4,  bearing  interest  from  date.  Say  the  date  of  the  settle- 
ment is  January  1,  1870.  Interest  on  $750  from  June  4,  1869^ 
to  January  1,  1870  (211  davs^  is  $26.37,  which,  added  ta 
$750=$776.37,  balance  due  with  interest,  January  1,  1870. 
This  result,  however,  may  be  ascertained  directly  by  the  In- 
ter est- Product  Method,  as  shown  by  the  same  example  on  page 
134. 


1 34  AVERAGE. 


INTEREST-PKODUCT  METHOD. 

Art.  10.  To  ascertain  the  balance  due  on  the  day 
of  settlement  the  Inter  est- Pro  duct  Method  may  be 
employed.  By  it  the  actual  date  of  settlement  is 
used  instead  of  an  assumed  date,  as  by  the  Discount- 
Product  Method. 

14.  Find  the  balance  due  January  1st,  1870,  of  the 
account  of  J.  (7.  Hintz,  page  133,  by  the  Interest-Prod- 
uct Method. 

1869 
Jnlv  3. ..$1000X182=  182000 
Jul>   7...     500X178=     89000 
Aug.  18...     250X136=    34000 


Au^.     1 $500X153=76500 

Aug.  13 500X141=70500 


Dr.  Interest-Product,  305000 
Cr.       "  "  147000 


Cr.  Interest-Product,  147000 


60)1580.00 


$  26.33    Interest  due  January  1,  1870. 
750.00    Balance  of  account. 

$776.33    Balance,  incUiding  interest. 

Explanation. — A  purchase  made  on  .July  3 — terms  cash — 
should  pay  interest  up  to  the  date  of  settlement.  That  being, 
in  tliis  case,  Jan.  1,  1870  the  debtor  is  to  be  charged  with  inter- 
est up  to  that  date,  viz.:  for  182  days.  The  interest  on  $1000 
for  182  days  is  equal  to  the  interest  on  S182,000  for  one  day, 
(or  on  $1  for  182000  days.)  The  same  with  every  item  on  the 
Dr.  side,  making  the  Dr.  Interest-Product  equal  to  the  interest 
on  $305,000  for  one  day.  The  day  of  settlement  being  Janu- 
arv  1,  1870,  Hintz  paid  $500  on  August  1,  1869,  153  davs  be- 
fore the  day  of  settlement,  and  $500  on  August  13,  1869,  141 
days  before  the  day  of  settlement,  and  is  therefore  entitled  to  a 
credit  for  interest  on  the  respective  amounts.  The  Cr.  Interest- 
Products,  147000,  being  equal  to  the  interest  on  $147,000  for  one 
day.  The  amount  on  which  Hintz  is  entitled  to  interest  for  one 
day  being $147,000,  and  the  amount  upon  which  he  is  chargeable 
with  interest  for  one  day  being  $305,000,  he  is  chargeable  with 
interest  on  $158,000  for  one  day  more  than  the  amount  upon 
which  he  is  entitled  to  receive  interest.     The  interest  on  $158,000 


AVERAGE.  135 


for  60  days  at  the  rate  of  6  per  cent  per  annum  is  $1580.00 — tliat 
being  1  per  cent  of  tlie  amount — 1  day  is  l-60th  of  60  days;, 
hence,  by  dividing  $1580.00  by  60  we  have  the  interest  for  1 
day  on  the  balance  of  the  products,  viz.:  $26.oo. 

The  advantage  of  this  method  over  the  Discount- 
Product  Method  is  obvious.  By  this  nothino;  is  as- 
sumed. Interest  is  actually  reckoned  from  the  date 
of  the  first  item  of  account  to  the  chiy  of  settlement^ 
and  the  accrued  interest  obtained  without  further 
calculation.  Should  the  timehe  required  it  is  readily 
found  by  dividing  the  difference  of  the  product  by 
the  balance  of  the  account.  By  the  Interest-Prod- 
uct Method  the  interest  is  simply  charged  to  the 
side  of  the  account  on  which  is  the  greater  pi*oduct, 
irrespective  of  the  bahmce  of  the  account;  whereas^ 
by  the  Discount-Product  Method  the  time  has  to  be 
reckoned  backward  or  forw^ard  from  the  date  ob- 
tained, and  the  interest  computed  and  applied  after- 
ward. 

15-16.  Find  the  balance  due  July  1st,  1877,  by  the 
Interest- Product  Method,  of  the  accounts  of  B.  H, 
Langdale  and  Edw.  Witte,  page  137. 

17-18.  Find  the  balance  due  January  1st,  1877,  by 
the  same  method,  of  the  account  of  N.  J.  Jones,  page 
137.     Also  of  G.  A.  Walworth's  account,  page  132. 

19.  July  1st,  1873.  K.  H.  Langdale's  account  is  a& 
follows : 

1873  1873 

Jan'y  3,  To  Mdse, $300.00  Feb.    6,  By  Cash, $    100.00 

"  4,    "       "      250.00  May   3,  "       "      1,000.00 

Feb.  9,    "      «      730.00  June  3,  "       "     160.00 

May  6,    "       "      800.00 

20.  Charge  him  with  the  interest  up  to  July  1^ 
1873,  close  his  account  and  bring  down  the  balance. 


136 


AVERAGE. 


Charge  him  with  goods  bought  since  the  day  of  last 
settlement,  as  follows:  July  3,  $500;  Aug.  6,  $100; 
Nov.  8,  $100.  Credit  him  with  cash,  paid  as  follows: 
Aug.  30,  $800;  Dec.  1,  $600.  Find  balance  due,  with 
interest,  January  1,  1874. 


COMPOUND  METHOD. 

Art.  11.  Find  the  balance  due^  July  1,  1874,  of  the 
following : 

21.    Theodore  Baur. 

1874  1874 

Jan.  12,  To  Cash, $  500.00    Jan.     3,  By  Mdse,  90  ds,...$1000.00 


Feb.    5,    "   Acc'p  60ds....    120.00    June  30,    ** 

Mar.    8,    "    Mdse, 600.00  * 

June  3,    "   Note,  3  mos...    100.00 

$1220.00 


60  ds,...  150.00 


$1150.00 


Arranged  according  to  the  dates  when  the  items 
are  due. 


Dr. 

Jan.     12 
Apr.       9 
Mar.       8 
Sept.      6 

Int. 

$500X170 

12()X  83 

500XU5 

luox 

Dis. 
67 

Cr. 

Apr.      3 
Aug.     29 

Dis. 

$1000X 
150X59 

Int. 
89 

Dr         Cr. 

85000 
99()0 
57500 

6700 

89000 

95700 

60)656.10 
$10.93  int.Dr 


8850 


Dr.  Ride,  of  ac.  $1220.00 
Cr.      ♦'  ♦*      1150  00 


Difference 70.00 

Interest  Dr 10.93 

Balance  due $80.93 


ExpiiANATiON.— It  will  be  seen 
that  both  the  Discount  and  In- 
terest-Product Methods  are  em- 
ployed in  tills  solution.  B  is  Dr. 
for  the  interest  on  the  three  first 
items  of  the  Dr.  side  of  his  ac- 
count, because  it  was  due  before 
the  day  of  settlement,  (July  1, 
'74),  and  he  is  credited  with  the 
discount  on  the  fourth  item,  that 
being  due  (Sept.  6,  74)  67  days 
after  the  settlement.  He  is  cred- 
ited with  the  interest  on  the 
$1000  paid  89  days  before  the  set- 
tlement, and  cUai'^ed  with  (he 
discount  on  the  $150  due  (Aug. 
29t  h,)  59  days  after  the  day  of  set- 
tlement; he  is  therefore  eh  a  ijre- 
able  with  the  interest  on  $1618.10 
for  one  day,  less  the  interest  on 
$957.00  for  that  time,  viz:  $10.93. 


COMPOUND  METHOD. 


137 


22-24.  Find  the  balance  due  January  1,  1877,  hy 
the  Compound  Method,  of  the  accounts  of  Langdale 
and  Witte,  Ex.  25  and  27,  also  the  balance  due  July  1, 
1876,  of  the  account  of  N.  J.  Jones,  Ex.  26. 

25.       E.  H.  Langdale. 


1»76 

1876 

July 

3,  ToMdse,  6mos 

,   560.87 

July 

1, 

By  Balance, 

127.15 

15,    '*       "     3    " 

149.50 

30, 

"  Accept.  60  ds,  300.00 

Aug. 

21,    "       •*      3    " 

2000.00 

Aug 

29, 

"   Cash, 

460.00 

Sep. 

18,    "        "      Cash, 

396.40 

Oct. 

20, 

*'  Note,  3  mos 

,  lUOU.OO 

Oct. 

15,    '*        •* 

175.20 

31. 

♦'  Cash, 

IdO.UO 

21     "        "           " 

425.16 

31, 

"  Mdse  Ret., 

250.00 

27,    "  Cash, 

100.00 

Nov. 

30, 

"  Cash, 

450.00 

31,    *•  Mdse,3mos 

,   506.18 

30, 

**  Balance, 

2144.77 

Nov. 

28,    «        "      4    ♦♦ 
30,    "        "      4    •* 

1,  To  Bal.. 

197.45 
321.16 

4^31.92. 

4831.92 

Dec. 

2144.77 

26. 

N.  1. 

Jones. 

1876 

1876 

Jau. 

1,  To  Balance, 

650.00 

Jan. 

8,  By  Mdse,  3  mos. 

160.00 

Feb. 

3,    "  Cash, 

245.00 

15,    * 

-       6    - 

710.87 

15,    ♦'  Note,  60  ds, 

416.87 

•  Feb. 

14      ' 

u       2    " 

910.14 

Mar. 

18,    "  Accept.  30  ds. 

1000.00 

Apr. 

16,    ' 

"     Cash, 

1000.00 

June 

4,    "       "          60    " 

750.14 

June 

8,    ' 

"       4  mos, 

9<i0.00 

16,    *'  Note,  3  mos, 

987.64 

15,    ' 

"       4    " 

2500.00 

80,    "  Cash, 

500.00 

17,    * 

«       6    '* 

121500 

30,    "  Mdse,  abate, 

200.00 

30,    * 

Sunds, 

700.00 

27.    Edward  Witte. 


1876 

1876 

July   3, 

To  Balance, 

1500.00 

Aug.   3,  By  Cash, 

1000.00 

18, 

•'  Mdse,  4  mos, 

750.40 

Sept.  7,    "    Accpt,  60  ds, 

600.00 

Aug.  29, 

"        "4    " 

128.80 

Nov.  5,    "        "        60  " 

7:)0.00 

Sep.  80, 

"        ««       4    u 

916.84 

Dec.  14,    "   Cash, 

2000.00 

Oct.   10, 

"        »       3    » 

500.00 

30, 

*.*  Cash, 

675.14 

Nov.  18, 

"  Sunds, 

564.18 

Answers:  $3253.15;  $2145.12;  $776.33;  $62.50; 
$2209.79;  $807.63;  $831.31;  $3355,76;  $848.11; 
$173.02;  $3252.93;  $3355.54:  $527.53. 


138  RATIO. 


XXIV.    RATIO. 

Art.  1.  The  relation  that!  one  number  hears  to  an- 
other is  called  ratio.  The  q^iotient  arising  from  di- 
viding one  number  by  another  of  the  same  denomi- 
nation, is  the  ratio  between-4hem! 


And  as  two  quotients  can  be  obtained  from  com- 
paring any  two  numbers,  it  follows  that  two  ratios 
can  also  be  obtained.  The  relation  tl%at  1  bears  to 
2  is  -|-,  and  that  which  2  bears  to'  1  is  f\ 

The  sign  of  ratio  is  the  colon.  \  TJ>e  above  ratios 
would  be  expressed  thus:  1:2  and  2:1,  and  would 
be  read  one  is  to  two  and  two  is  to  one.  French 
mathematicians  divide  the  first  term  by  the  second; 
English  the  second  by  the  first.     The  English  method 


is  used  here,  3  :  6  will  equal  |-  or  2,     ^  :  |-=-=::r|. 


Art.  2.  Numbers  or  quantities  of  different  de- 
nominlations/  can  not  have  a  ratio.  lYe  can  not 
compare  3  trees  with  5  books.  But.if  the  numbers 
are  capable  of  being  reduced  to  the  same  denomina- 
tion, they  can  be  compared ;  for  we  can  say  3  feet 
is  to  2  inches,  as  it  is  the  same  as  to  say,  36  inches 
is  to  2  inches.. 

Each  number  is  called  a  term  of  the  ratio.  The 
first  term  is  called  antecedent;  the  second,  conse- 
quent. 

The  value  of  a  ratio  depends  upon  the  relative 
size  of  its  terms. 

Every  ratio  may  be  formed  into  a  fraction  by  mak- 
ing the  consequent  the  numerator  and  the  antecedent 
the  denominator,  thus:    4:8  =  |  =  2;  and  8:4  =  | 


PROPORTION.  139 


XXV.    PROPORTION. 

Art.  1.  Two  ratios  may  be  equal  to  each  other. 
2  :  4,  =  4  :  8. 

2  bears  the  same  relation  to  4  that  4  does  to  8. 

Art.  2.  When  ratios  are  equal,  the  numbers  or 
terms  which  compose  them,  are  said  to  be  in  propor- 
tioUj  and  are  written  thus:  2  .  4  :  :  3  :  6,  and  read  2  is 
to  4  as  3  is  to  6. 

The  first  and  last  terms,  as  the  2  and  6,  are  called 
extremes,  and  the  second  and  third  the  means. 

Art.  3.  The  same  ratio  may  arise  by  comparing 
4  quantities,  two  of  which  are  different  in  denomina- 
tion from  the  other  two. 

tuns      tuns  $         $ 

3     :     6     : :     6  :  12.     The  ratio  is  2. 

Art.  4.  *  If  the  extremes  are  multiplied  together, 
the  product  will  be  equal  to  the  product  of  the  means, 

3X12=36 
6X   6=36 

Hence,  when  any  3  terms  are  given,  we  can  readily 
find  the  fourth,  by  dividing  the  product  by  the  odd 
term.     If  we  had  only  the  three  first  terms  of  the 

tuns    tuns  $ 

above  proportion:    that  is,  3    :  6    :  :    6,  the    fourth 
term  would  be  found  by  dividing  the  product  of  6X6, 

or  36  by  3,=:12,  or  the  fourth  term  as  above. 

To  apply  this  in  practice,  we  have  only  to  sup- 
pose the  3  tuns  and  6  tuns  to  be  coal,  and  the  $6,  the 
price  of  3  tuns.  Then  3  tuns  is  to  6  tuns,  as  the 
price  of  3  tuns  is  to  the  price  of  6  tuns. 


140  PROPORTION. 


2.  What  will  35  lbs.  of  sugar  cost,  if  7  lbs.  cost  77 
cents  ? 

Statement. — 7  :  35  : :  77  is  to  the  price  of  35. 

lbs.     lbs.      cents. 
7  :  35  : :  77 
35 

385 

231 

7)2695 

$3785 
The  above  operation  might  have  been  abridged  by 
cancellation. 

H   u   ^ 

77X5=3.85  ^n5. 

3.  If  6    lbs.   of  tea  cost  $4.75,  what  will  15   lbs. 
cost  ? 

4.  Find  the  price  of  37  horses,  when  16  cost  $1500? 

5.  What  will  120  bbls.  of  potatoes  cost,  if  21  cost 
$67? 

6.  42  bushels  of  beans  cost  $87.50,   what  will  3 
bushels  cost  ? 

7.  If  610  bushels  of  wheat  cost  $1670,  what  will 
be  the  price  of  27  ? 

8.  If  3  bushels  1^  pecks  of  beans  cost  $12.50,  what 
will  5  bushels  4  qts.  cost? 


bu.    pks. 

bu. 

qts. 

The    bushels, 

Statement, — 3    1^ 

:     5 

4: 

:  12.50 

pks.,  and  qts. 

4 

4 

had   to  be  re- 

duced  to  qts.. 

13 

20 

in  order  to  be 

8 

8 

of    the    same 

denomination. 

108 

164X12.50 

Art.  176. 

• 

— = 

=$18.98 

\ 

108 
iin5M?(?r5.— $3468.75,  $11.87*,  $6.25,  $382.86,  $73.92: 

$.185. 


PROPORTION.  141 


9.  If  27^  lbs.  of  butter  cost  $3.75,  what  will  16^ 
lbs.  cost? 

10.  Find  the  price  of  12^  dozen  of  chickens  at  30 
cents  a  pair. 

11.  The  price  of  21  tuns,  13  cwt.,  3  qrs.,  and  15 
lbs.  of  hemp  is  $1680.55,  what  will  15  cwt.  cost? 

12.  What  will  54  lbs.  7^  oz.  of  tea  cost,  if  15J  lbs. 

cost  $8.47? 

13.  If  f  of  a  ship  cost  $7000,  what  will  j%  cost? 

These  fractions  need  not  be  reduced  to  the  same 
denomination. 

14.  If  6  men  do  a  piece  of  work  in  7  days,  how 
long  will  it  take  5  men  to  do  it  ? 

In  stating  the  previous  question,  we  compared  quan- 
tity with  quantity  and  cost  with  cost.  In  this  question 
there  is  nothing  relating  to  cost,  so  we  must  adopt  another 
method  of  making  the  statement.  Perhaps  the  simplest 
is  the  following  : 

1.  Inquire  what  is  wanted,  and  put  the  term  of  that  name  to 
the  right.  In  the  question  above,  time  is  wanted,  so  we  put  the 
term  of  that  name  to  the  right. 

2.  Ascertain  by  reasoning,  whether  the  quantity  wanted  will 
be  greater  or  less  than  that  given ;  if  less,  put  the  smaller  of  the 
two  numbers  for  the  middle  term ;  if  greater,  put  the  greater  of 
the  two  terms  for  the  micMle  term.  In  the  above  question,  we 
reason,  that  it  will  take  5  men  a  greater  time  than  6  men,  so  we 
put  the  greater  of  the  two  terms  (6)  in  the  second  place. 

men.     men. 

Statement. — 5  :  6  :  :  7  days.  The  answer  is  8| 
dpys,  or  8  days  4  hours,  reckoning  10  hours  to  the 
d^y. 

15.  If  2  men  plow  a  field  in  3  days,  how  long  will 
^*    take  3  men  to  do  it? 

Answers,— i22.b0,  $2.25,  $7350, $58.09, $30.25,  2. 


142  PROPORTION. 


16.  If  26  yards  of  linen  cost  $13.50,  what  will  10 
yards  cost? 

17.  If  8  coats  can  bo  made  from  10-1  yards  of  cloth, 
how  man}^  can  be  made  from  31^  yards? 

18.  If  the  interest  of  S750  for  3  years,  4  months, 
and  10  days  be  $151.25  (360  days  to  the  year),  w^hat 
is  it  for  one  year? 

19.  The  interest  of  £100,  from  3d  of  April  to  25th 
February,  is  £6  5s.  9'?Jd.,  what  is  it  per  year? 

20.  A,  E,  and  C  are  in  partnership,  and  their  gains 
for  the  year  are  $6757,  what  is  each  man's  share, 
suppose  A  invested  $1567,  B  $2600,  and  C  $3798? 

The  sum  of  their  investments  is  to  each  man's  in- 
vestment, as  the  total  gains  to  each  man's  gain. 

21.  M  invests  $6500,  N  $1487,  O  $3654;  in  three 
months,  it  is  found  that  their  gains  are  $1678,  w^hat 
IS  each  man's  share? 

22.  A  lends  B  $1000  for  13  months  10  days,  how 
long  should  B   lend   A  $8271,  to  return  the  favor. 

23.  If  the  shadow  from  a  two  foot  rule  be  6  in., 
what  is  the  hight  of  the  tree  that  throws  a  shadow 
of  75  feet  ? 

24.  If  7  men  can  hipld  21  perches  of  masonry  in  a 
day,  how  many  men  will  it  require  to  build  156  perches 
in  a  day. 

25.  The  shadow  of  a  tree  being  87  feet;  two  nails 
being  driven  in  the  tree  3  feet  apart,  show  a  distance  on 
the  tree  of  4J  feet,  what  is  the  height  of  the  tree? 

26.  The  net  profits  of  a  concern  being  $1860;  A's 
interest  is  $8750,  and  B's  interest  is  $8190 ;  what  is 
each  man's  gain  ? 

Answers:  9,  7,  300,  58,  52,  49,  $5.19,  $5.90,  $4.50 
$45,  $910,  $936.89,  $950,  $1329.34,  $936.94. 


PARTNERSHIP.  143 


XXVI.    PARTNERSHIP. 

Art.  1.  When  two  or  more  persons  associate 
together  to  carry  on  a  business,  they  are  said  to  be 
in  partnership,  and  are  called  a  firm^  house,  or  com- 
pany. 

The  funds,  propert}^,  and  merchandise  furnished 
by  partners  for  carrying  on  business,  are  called 
Btock  or  capital,  and  the  gains  are  called  dividends. 

The  liabilities  of  a  partnership  or  individual  busi- 
ness are  the  debts,  and  the  assets  their  available 
means,  including  the  indebtedness  of  others  to  them. 

An  inventory  is  a  list  or  statement  of  those  things 
which  constitute  assets. 

Art.  2.  In  keeping  partnership  accounts,  each 
member  of  the  firm  should  be  credited  with  all  that 
he  brings  into  the  concern  or  business,  and  be  charged 
or  debited  with  all  he  takes  out,  just  the  same  as  if 
he  had  no  interest  in  it. 

Art.  3.  The  calculations  peculiar  to  partnership, 
relate  to  the  division  of  property  and  profits. 

1.  A,  B,  and  C  have  been  in  business  one  year,  and 
find  they  have  made  a  net  gain  of  $3476,  which  is  to 
be  divided  as  follows :  A  is  to  have  J,  B  J,  and  C  J; 
required  the  share  of  each. 

$3176^^1738,  A's  share;  $3  4_7  6=$869=B^8  share; 
and  $869=C'8  share. 

2.  X,  Y,  and  Z  purchase  a  tract  of  land  for  $2000; 
X  giving  $600,  Y  $900,  and  Z  the  remainder.  In  one 
year  afterward,  they  sell  it  for  $5500;  required  each 
person's  share  of  the  proceeds. 

3  A,  B,  and  C  invest  $2000  each.  In  3  months  their 
gross  gains  are  $2000  ;  expenses,  including  $250  for 
additional  services  of  C,  $600,  what  will  be  each  man's 
share  of  the  gain? 


144  *     PARTNERSHIP. 


4.  D's  interest  in  a  partnership  is  ■^.  What  is  his 
share  of  a  gain  to  the  firm  of  $3467.18  ? 

5.  E,  F  and  G  own  a  steamboat  worth  $35,000,  their 
respective  shares  being  ^,  -^,  -^q.  What  is  the  profit  of 
each  after  deducting  $1350  expenses,  from  $5450.  gross 
profits  ? 

Art.  4.  Interest  on  Investment. 

6.  H,  I  and  J  invest  in  partnership  $3400,  $2900,  and 
$1500  respectively,  and  at  the  end  of  the  year  find  a  net 
gain  of  $2600.  Allowing  6%  on  their  investments,  what 
amount  is  each  entitled  to  in  proportion  to  the  capital 
advanced  ? 

Interest  on  Investment,  8468. 

Net  profits,  $2600,  minus  $=468=^2132,  to  be  divided  pro  rata. 
$2132  :  7800=.2733  gain  on  :ffl.OO. 

.2783X3400=$929;33  H's  share. 

.2733-X2900=  792.67  I's  share. 

.2733-X1500=  410.00  J's  share. 


82132.00,  whole  gain. 
The  respective  shares  of  gain  may  be  ascertained  bytlie  follow- 
ing proportion:  The  whole  investment  is  to  H's  investment  as 
the  wliole  gain  is  to  H's  gain,  thus: 

87800  :  3400  :  :  2132  :  H's  gain. 
39       17      X  2132 
39)36244 


#929.33 
$7809  :  2900  :  :  2132  :  792.67  (I's  gain) 
8VS00  :  1500  :  :  2132  :  410.00  (J's  gain) 

7.  K,  L  and  M  engage  in  partnership  with  a  capital  of 
$15000,  to  share  equally,  K  investing  $10000,  L  $3000, 
and  M  $2000,  and  L  and  M  to  receive  salaries  of  $1500 
and  $1200  a  year  respectively ;  allowing  interest  on  their 
investments,  which  remained  intact,  what  is  each  part- 
ner's share  in  a  gross  gain  of  $5700,  expenses  being 
$1950,  exclusive  of  partner's  salaries? 

Art.  5.    Winding  up  a  Losing  Concern, 

8.  R,  S  and  T,  equal  partners,  with  a  capital  of  $30000, 
finding  that  they  are  losing  money,  agree  to  dissolve,  and 
on  March  4,  1874,  leave  the  property  in  the  hands  of  T 
to  settle.     At  this  time  the  effects  were  cash  on  hand 


PARTNERSHIP.  145 

$500,  merchandise  $17500,  bills  receivable  $1300,  and 
book  accounts  $1000,  and  their  liabilities  were  bills  pay- 
able to  the  amount  of  $2100.  On  September  first  T 
reports  as  follows  :  sales  of  merchandise  $14000,  on  hand 
$1500,  cash  on  hand  $13000,  notes  $300,  uncollected  bills 
$750,  liabilities  extinguished ;  expenses  $650.  Of  the 
remaining  efiects  T  proposes  to  take  the  Mdse  at  a  dis- 
count of  50%  if  his  partners  take  notes  and  accounts 
at  the  same  rate.  Failmg  to  agree,  they  sell  the  goods 
at  auction  for  $900,  and  T  agrees  to  take  the  bills  re- 
ceivable in  payment  for  the  collection  of  the  unsettled 
bills  which  he  thus  guarantees.  Required  the  amount 
coming  to  each,  allowing  T  1  %  commission  for  settling 
the  business? 

Art.  6.  Average  Capital, 

9.  U,  V  and  W  engage  in  business  January  1,  1874, 
investing  respectively  $3000,  $2000,  and  $1000,  and 
agreeing  to  share  the  gains  and  bear  the  losses  in  the 
ratio  of  their  average  capital.  A-pril  first  U  draws  $100, 
May  first  V  draws  $200,  and  July  first  W  draws  $100. 
Assuming  the  gains  to  be  $1500  at  the  end  of  the  year, 
what  was  each  partner's  share  ? 


u. 

Int.  on  ^3000  for  12  mos.  $180 

$175.50 

•♦     "       100    "    9    "            4.50 

112.00 
57.00 

$175.50 

V. 

$344.50 

Int.  on  $2000  for  12  mos.  8J 20.00 

344.50 

:  175.50  :  :  1500=U's  share. 

"      "       200  "      8    "           8.00 

:  112  00  : :  1500= V's      " 

:  57.00  ::  1500=W's    *' 

W.                          $112.00 

Int.  on  $1000  for  12  mos.  mo.oO 

"     "       100   "    6     "          3.00 

$57.00 

The  question  may  also  be  solved  by  Products. 

Answers:  $768.75,  $1281,25,  $5281,00,  $1083,49,  $4834.50,  $1738,  $860, 
$487,66,  $764.15,  $248.19,  $2050,  $466.66,  $1650,  $1375,  $2475,  $4100,  $50. 

10 


146  CORPOKATIONS. 


XXVII.  JOINT  STOCK  COMPANIES. 

Art.  1.  A  JoiJit  Stock  Company  is  a  body  of  men 
associated  together  in  a  species  of  partnership,  to 
carr}^  out  some  heavy  undertaking  requiring  the  in- 
vestment of  more  capital  than  individuals  or  part- 
nership companies  commonly  possess.  Joint  stock 
companies  are  usually  incorporated  by  act  of  legis- 
lature, with  certain  privileges.  Eailroads,  canals, 
bridges,  etc.,  are  generall}'  constructed  bj^this  species 
of  combined  interest,  and  many  banking  and  insur- 
ance houses,  scholastic  institutions,  etc.,  are  owned 
and  managed  by  joint  stock  companies. 

When  an  association  of  this  kind  is  to  be  formed, 
a  few  leading  persons  make  an  estimate  of  the  prob- 
able amount  of  capital  required,  divide  it  into  equal 
shares  of  from  $10  to  $100,  or  $500,  according  to  the 
nature  of  the  undertaking,  and  issue  certificates  of 
ownership  for  each  share.  These  are  called  certifi- 
cates of  stocky  and  are  transferable.  Persons  own- 
ing certificates,  are  called  stockholders. 

Joint  stock  companies  are  usually  managed  by  a 
president  and  board  of  directors,  elected  for  the  pur- 
pose, by  the  stockholders. 

When  shares  sell  for  the  price  named  in  the  certi- 
ficate, the  stock  is  said  to  be  at  par;  if  above  this 
value,  they  are  said  to  be  above  par;  if  below  it, 
below  par. 

Besides  the  stocks  of  companies,  there  are  govern- 
ment stocks,  which  consist  of  bonds  that  have  been 
issued  by  state  officers,  for  the  purpose  of  borrowing 
money.     These  draw  interest  at  a  specified  rate. 

In  dividing  the  profits  of  joint  stock  companies,  it 
has  been  found  more  convenient  to  declare  the  divi- 
dend as  so  much  per  cent. 


COEPORATIONS.  147 


1.  What  is  the  cost  of  10  shares  of  railroad  stock 
at  5  %  below  par,  the  original  cost  being  $100  per 
share? 

Find  the  cost  of  10  shares,  at  SlOO  and  deduct  5  ^. 

2.  A  banking  institution  declares  a  dividend  of 
18  %  on  a  capital  of  $376198,  what  amount  of  money 
should  a  stockholder  receive,  who  holds  5  shares 
valued  at  $200  each? 

3.  I  hold  15  shares  (each  of  $100)  of  stock,  in 
gas  works,  which  have  declared  a  dividend  of  20  %, 

"  how  much  am  I  entitled  to  after  my  gas  bill  of  $20, 
is  deducted  ? 

4.  How  many  shares  of  United  States  stocks  at 
2  %  above  par,  can  I  biiy  for  $1224,  the  original 
cost  being  $100  per  share? 

5.  What  amount  of  stock  can  I  buy  for  $1687,  if 
I  am  allowed  2  %  commission  on  the  amount  in- 
vested ? 

The  amount  I  am  to  receive  is  to  be  j§^,  or  ^'^  of 
the  amount  of  stock  purchased — not  J-^  of  $1687,  for 
that  would  be  commission  on  commission  and  invest- 
ment. 

Let  the  amount  to  be  invested  be  represented  by  f{{, 
and  to  this  add  -'^=f  J  ;  then  we  discover  that  $1687 
is  1^  of  the  amount  to  be  invested,  '||'''=33.078= 
g?^,  or  my  commission,  which  if  we  multiply  by  50, 
will  give  us  the  amount  to  be  spent,  $1653.90. 

To  prove  this,  find  the  com.  on  $1653.90  at  2  %. 

6.  A  broker  receives  S6785,  which  he  is  desired  tc 
invest  in  State  stocks,  how  much  should  he  invest, 
and  allow  himself  2|  ^  on  the  investment? 

7.  What  amount  of  stock  can  a  broker  buy  for 
16700,  and  allow  himself  J  %  on  the  investment? 

Answers:  $180,  $950,  $280,  $16658.35,  $6619.49, 
12,  $1780. 


148  COMPOUND   NUMBERS. 


XXVIII.  COMPOUND  NUMBERS. 

The  application  of  the  fundamental  rules  to  num- 
bers of  different  denominations,  gallons,  quarts,  and 
pints;  hundreds,  quarters,  and  pounds^  etc.,  will,  it  is 
presumed,  be  sufficiently  taught  in  the  following 
examples : 

BRITISH   MONEY. 

Art.    1.     Tg  add  compound  numbers. 

What  is  the  amount  of  the  following  sums  of  Brit- 
ish money? 

Solution. — We  first  add  the  fractions,  calling 
£.  S  d  thcn^  bX\  farthings^  which  makes  6  farthings ;  these 
1ft      17      4-1      ^®  reduce  to  pence,  by  dividing  them  by  4.     |=r 

1Q        fi      ?!      ^1^^  ^**     ^"^  *'  ^^^  ^^^  *^®  ^  P®^"^  ^^  *^^® 
n        ^7      of      column  of  pence,  which  makes  20  pence;  this  num- 

17        7      oj     ber  divided  by  12  (the  number  of  pence  in  a  shil- 

rr      TZ      ^     ling)=l  shilling  and  8  pence.    Write  the  8  under 

Q'i)      ii      o-^     ^jjg  pence,  and  add  1  to  the  units  of  the  shilling's 

place,   which  makes  21 ;  write   1,  and  add  the  2 

to  the  ten's  column  =3  or  31  shillings,  which  divided  by  20=£1 

and  11  shillings  left.     Write  the  latter  under  the  shillings,  and 

add  the  1  pound  to  the  pound's  column  =<£55.   Ans.  £55,  lis.  8^d. 

2.  Add  the  following: 

£  17  18  ll|+£  14  17  2i+£  16  14  8  = 
£  17  19  0J+£  45  0  llf +£111  10  21= 
£116  16    6   -l-£320  14    5|+£  38  18  8  = 

Total,  £700,     O5.    M. 
Art.   2.     To  subtract  compound  numbers. 
Subtract  £14,  75.  Q\d.  from  £19,  4s.  M. 

£       5,     d.  Solution. — We  can  not  take  J  from  nothing, 

29      4      3  so  we  add  a  penny  to  both  numbers;    then  sub- 

14      7      64-  tracting  the  1  from  a  penny,  or  i,  we  have  | 

^  left.     Adding  Id.  to  the  6d.,  we  have  7d.,  which 

A    -tr*      Q  3  ^®   ^^^  ^^^  subtract  from  the  3d.  above,  and 

*   ^^      ^i  accordingly  add  Is.  to  both  numbers;  7  from 


COMPOUND    NUMBERS.  149 

Is.  3d.  or  15d.,  leaves  8d.  Adding  Is.  to  the  shillings,  we  have 
8s.,  which  can  liot  be  taken  from  4s.  without  adding  £1  to  both 
numbers;  £1  to  4s.=24s. ;  8s.  from  24s.=16s.  Then  adding  £1 
to  the  14,  we  have  £15,  which,  taken  from  £19^=£4,  making  the 
answer  £4  16s.  8|d. 

2   Subtract  the  following  : 

£      s.  d.        £    s.  d. 

17  10  Si—   14  5  3  = 

119  7  6  —  17  19  5J= 

500  0  0  —  20  18  8  = 

176  14  7J— 129  15  7i  = 


Total,  £620,  13s.  9^^. 

Art.  3.     To  multiply  compound  numbers. 

Multiply  £17,  4s.  d^d.  by  8. 

Operation.    £17    4       9  J 
8 


£137  18.      2. 

After  performing  operations  in  addition,  the  learner 
will  readily  see  how  this  is  done. 


£    s. 

d. 

17  18 

8JX   7  = 

120  16 

6^X12= 

365     0 

7JX   9= 

Total,  £4860,  15s.  OJd 
Art.    4.    To  divide  compound  numbers. 

Divide    £157,   13s.  6}^.    equally  between  25   per 
sons: 


150  COMPOUND  NUMBERS. 


Operation.     25)£157  135.  6^  (£6,  6s.  l^d.,  or 

150  £6,  65.  If6?.,  nearly. 

£7= remainder. 
20 


153=shilHngs  in  £7,  with  135.  of  the 
150  [dividend  added. 

3=remainder  in  shillings. 
12 


42=pence  in  3 shillings,  and  6  pence 
25  [from  the  dividend. 

17=remainder  in  pence. 
4 

70=farthing8  in  17  pence  and  J. 
50 


20=remainder,  or  |g  farthings. 

Recapitulation. — We  first  divided  the  £157  by  25;  then  163 
shillings  by  25 ;  then  42  pence  by  25 ;  and,  lastly,  the  70 
farthings. 

£     s.     d. 
Divide  167  18  6f  by  25  = 
768  14  3}  by  125= 
17  11  3J  by  875  = 

Total,  £12^"l75^ 

£     5.  d. 

Divide       25  18  4  by     5 

76  12  8  by     4 

1  15  9  by     3 

162  12  6  by  30 


Total,  £30  75.  2d. 


FOREIGN    EXCHANGE.  151 


XXIX.   "FOREIGN  EXCHANGE. 

Art.  1.  In  calculating  Foreign  Exchange  the  money 
of  one  country  has  to  be  expressed  in  that  of  another. 
A  bill  drawn  in  New  York  on  an  English  house,  will  be 
expressed  in  pounds,  shillings,  and  pence, ' 

The  relative  value  of  moneys  of  different  countries 
depends  on  the  par  of  ExcJiange,  and  the  course  of  Ex- 
change. 

The  Par  of  Exchange  is  the  comparative  value  of  the 
coins  of  the  different  countries,  and  is  fixed,  while  the 
relative  purity  of  the  coins  is  the  same.  The  par  of 
exchange  between  the  United  States  and  Great  Britain 
is  $4.8665  to  the  pound  sterling. 

The  Course  of  Exchange  usually  depends  upon  the 
relative  state  of  indebtedness  of  the  merchants  of  the 
different  countries,  and  the  supply  of  gold  and  silver; 
accordingly,  the  course  of  exchange  will  sometimes  be 
above,  and  sometimes  below  par. 

FORM  OF  A  FOREIGN  BILL. 

Exchange  for  £1567.  Cincinnati,  June  3, 1876. 
Thirty  days  after  sight  of  this  first  of  Exchange 
(second  and  third  of  the  same  tenor  and  date  unpaid,) 
pay  to  the  order  of  J.  H.  Story,  the  sum  of  one  thousand 
hwQ  hundred  and  sixty-seven  pounds  sterling,  value  re- 
ceived, and  place  .to  my  account  as  advised. 

To  William  Morgan',  Esq.,  C.  H.  GuiOU. 

Liverpool,  England. 

Note. — Foreign  Bills  are  generally  drawn  in  seteoftwo,  three, 
or  four ;  that  is,  copies  of  the  same  bill  are  made  out  and  trans- 
mitted by  different  conveyances  to  the  payee,  one  of  which 
being  received  and  accepted,  or  paid,  the  others  to  be  void. 


152 


FOREIGN   EXCHANGE. 


BKITISH  OR  STERLING  EXCHANGE. 

Aet.  2.  British  or  Sterling  Money  Reduced  to  Federal 
Money  or  United  States  Curi'ency. 

The  calculations  relating  to  sterling  money  have  been 
reduced  to  simple  operations.  In  the  daily  papers  we  find 
quoted  in  gold  or  currency  the  precise  value  of  the  pound 
sterling  in  dollars  and  cents,  as  in  the  following  example, 
the  operation  of  which  we  give  below. 

1.  Required  the  value  of  £157  9  2  in  Federal  Money, 
when  sterling  exchange  is  quoted  at  4  86  in  gold,  and 
gold  at  10%  or  110. 

By  Decimals. 

12)2.0 
20)9.166 
£157.4583 
486 
9447498 
12596664 
6298832 

^765.247338==cost  in  gold. 


By  Aliquots. 

486 
157  9  2 


3402 
2430 

486 


s.d. 

76302 

6  8= 

=\ 

162 

2  6= 

-i 

607 

765.247. 
76.524: 


=cost  in  gold. 
-10% 


|841.771=cost  in  currency. 
Akt.  3.  To  assid  the  learner  we  give  ihefoUovdng  Table  of 

AlIQUOTS  OF  A   POUND. 


10   o=i 

6    8=.i 
5    0=i 


K  d. 

4  0= 

3  4= 

2  6= 


d. 
0-tV 


d. 
6-^ 


2.  Sterling  at  4  87J  in  gold,  and  gold  at  llOJ, 
quired  the  currency  for  £147  6  8. 


re- 


FOREIGN   EXCHANGE.  153 

3.  The  quotation  for  sterling  being  540  in  currency, 
how  much  will  buy  a  bill  for  £652  10? 

4.  Required  the  currency  for  the  following  l)ill  at  3 
davs  sight ;  486J^,  gold  at  110 ;  £376  4  6. 

5.  What  will  pay  for  a  sight  bill  for  £319  4  9,  with 
the  market  at  489  in  gold,  and  gold  at  9 J  premium  ? 

6.  How  much  will  a  bill  for  £794  5  4,  cost  in  cur- 
rency,  sterling  exchange  being  quoted  at  486  i,  and  gold 
at  110? 

7.  Required  the  cost  of  £113  3  3  at  the  same  quota- 
tions.    (Ex.  6.) 

Art.  4.     To  Reduce  Federal  to  Sterling  Money. 

8.  How  much  British  Money  can  be  bought  for 
$841.77,  exchange  being  quoted  at  486  in  gold,  and 
gold  at  110? 

In  other  words,  if  £1  cost  $4.86+10%,  what  sum  in 
the  same  currency  can  be  bought  for  S841.77? 
841.77 :  4.86+.486=157.458 

20 

9.160 
12 


1.920^£157  9  2.    (See  Ex.1.) 

9.  Sterling  at  4  89  in  gold,  and  gold  at  107J,  what 
amount  of  a  bill  can  be  bought  for  $1051.35? 

10.  Required  the  amount  of  a  bill  that  can  be  bought 
for  $31.49,  sterling  quotations  being  4  86f ,  in  gold,  and 
gold  being  111^. 

11.  Sterling  quotations  being  520,  in  currency,  what 
sight  bills  can  be  bought  for  $650  ? 

12.  Required  the  face  of  a  sight  bill  that  can  be 
bought  for  $50,  sterling  quotations  being  489  in  gold, 
and  gold  at  10%  premium. 

13.  Sterling  at  487 J  in  gold,  and  gold  at  109,  what 
amount  of  a  bill  can  be  bought  for  $79.56? 

14.  Required  the  amount  of  a  bill  that  can  be  bought 
for  $47.20  currency,  sterling  at  487,  gold  at  108? 


154  FOREIGN    EXCHANGE. 


Answers:  $841J7,  $1709.37,  $791.87,  $3523.50, 
$2013.37,  $4250.52,  S605.58,  £157  9  2,  £200,  £14  19  5, 
£5  16,  £8  19  4,  £1  10,  £125,  £246  13,  4.  £9  5  11. 

GERMAN  EXCHANGE. 

Art.  5.  The  money  of  the  whole  German  Empire 
is  Beichmark  and  pfennige. 

Signs:  Rm.  Reichmark,  d.  pfennig.     1  i?m.=100c?. 

COMPARATIVE   TABLE. 


Rm. 
1  Prussian  Thaler  (30,Silber- 

groschen,)  ...  =3 
1  Florin,  Austrian  Coinage,  =  2 
7  Florins,  S'th  Qer.  Currency, 

(Siiddeutsche   Wahrung,)  =12 


Rm 

10  Mark-Banco,  (Hamburg,)=  15 

100  Florins,  (Holland,)        .    =169 

5  Francs,  (France,)        .       =    4 

10  £,  (British,)        .        .        .  =203 

"--^      ■      'Tede 


5  Francs,  (France,)  .  =  4 
0£,  (British,)  .  .  .  =203 
97  Cents,  (Federal  Money,)  =    4 

Art.  6.   To  Reduce  German  to  Federal  Money. 

1.  Required  the  value  of  Rm,  1264  in  United  States 
currency  when  exchange  is  at  par,  and  gold  is  quoted 
at  110. 

First  Method.  Second  Method. 

1264                           Rm.  A  :  Rm,  ItU  :  :  97  c  :  a; 

24.25                             1  316 

■316^  ?1 

5056  2212 

2528  2844 

$306.52  in  gold.  $306.52  in  gold. 

30.65^10%  cost  of  gold.  30.65 

$337.17  in  currency.  $337.17  in  currency. 

Explanation — Fir^t  Method. — If  one  Rm.  is  equal  to  24Jc 
in  gold,  Rm.  1264  are  equal  to  1264  times  24^c,  viz.:  $305.62  in 
gold.  Gold  being  quoted  at  110;  i.e.,  10  per  cent  above  par, 
we  add  10  per  cent  to  the  value  in  gold  to  obtain  the  value  in 
currency. 

The  Second  Method  is  by  proportion,  which  see  page  139. 

2.  The  quotations  for  German  Exchange  being  98^, 
and  for  gold  110^,  find  the  cost  in  currency  of  a  bill  for 
Rm,  1892. 


FOREIGN   EXCHANGE.  155 

3.  German  Exchange  at  100,  and  gold  at  llOJ,  what 
will  be  the  cost  of  a  bill  for  Em.  720  ? 

4.  What  will  be  the  cost  of  a  bill  for  Em.  58,  German 
exchange  being  quoted  at  98,  and  gold  at  112^. 

Art.  7.   To  Reduce  federal  to  German  Money. 

How  much  German  Exchange  can  be  bought  for 
'^125.40  in  currency,  when  the  quotation  is  4:96   and 

gold  no? 

24  Explanation. — If  four  B^n.  are 

2.4  equal    to  96c  in   gold,  one  Rm.  ia 

26!4J1 2540.0(475  Bvi.  equal  to  24c  in  gold,  gold  being  at 

IQ5g  10  per  cent  premium,  one  Bm.  will 

-.Q^^  cost   10   per  cent  more  in  currency 

!.Zf^  or  26  4.10th  cents.     $125.40  in  cur- 
rency will  bring  as  many  Bm.  as  26.4 


1^2^  is  contained  therein,  viz :  Bin.  475. 

i:j20 


5.  Required  the  amount  of  a  bill  that  can  be  bought 
for  $1862,  exchange  being  quoted  at  95,  to  Bm.  4,  and 
gold  at  112? 

6.  German  Exchange  at  96,  and  gold  at  110,  what 
amount  of  a  bill  can  be  bought  for  $125  ? 

7.  Required  the  amount  of  a  bill  that  can  be  bought 
for  $42.25  currency,  exchange  being  quoted  at  96  and 
gold  at  111? 

8.  What  is  the  face  of  a  bill  that  costs  $666.68  in 
currency,  exchange  100,  gold  115? 

Answers:  Rm.  475,  Rm.  473.45,  Rm.  7000,  Rm. 
158.60,  $15.98,  $512.86,  $198.90,  $337.17,  Rm.  2318.89. 

FRENCH  EXCHANGE. 

Art.  8.  The  unit  of  French  Money  is  the  franc,  (a 
silver  coin  equal  in  quality  to  our  silver  coins.)  l/c.= 
10  decwies,  1  (/ec.=10  centimes. 

The  par  value  of  the  fc.  is  about  19|^  cents,  or  fcs.  512^ 
to  $100  in  gold. 

Art.  9.   To  Reduce  French  to  Federal  Money. 

1.  French  Exchange  being  quoted  at  510  and  gold  at 


156  FOREIGN   EXCHANGE. 

Ill,  requii:ed  the  cost  in  United  States  currency  of  a 
bill  for /cs.  2465. 

510)2465.00(4.83-  4.83- 

2040  111 

4250  4.83- 

4080  48.33- 

1700"  483.33- 

1530  S536.50  in  U.  S.  currency. 

It  will  be  noticed  that  the  amount  of  the  bill  is  divided  by 
the  quotation  of  the  French  Exchange,  and  the  result  multi- 
plied by  the  gold  quotation. 

French  Exchange  may  be  worked  by  proportion. 
/cs.510:/cs.  2465^$111  :  $536.50. 

2.  Kequired  the  cost  in  currency  of  a  bill  for  /cs.  727.6, 
exchancre  being  quoted  at  520,  and  gold  at  110. 

3.  What  will  be  the  cost  of  a  bill  for  fcs.  226.66, 
French  Exchange  518,  gold  112^? 

4.  French  Exchange  at  518 J,  and  gold  at  115,  what 
will  be  the  cost  of  a  bill  for/cs.  52.5? 

Art.  10.   To  Reduce  Federal  to  French  Money, 

5.  The  quotation  being  510  for  French  Exchange  and 
111  for  gold,  required  the  amount  of  a  bill  that  can  be 
bought  for  $536.50  currency. 

The  process  is  the  reverse  of  the  above.  The  propor- 
tion would  be  $111  :  $536.50  :  :fc8.  510  :  fcs.  2465. 

6.  Kequired  the  amount  of  a  bill  that  can  be  bought 
for  $260  currency,  French  Exchange  being  quoted  at 
515,  and  gold  at  110. 

7.  French  Exchange  at  516|,  and  gold  at  109|,  what 
amount  of  a  bill  can  be  bought  for  $1410  currency? 

8.  The  quotation  being  512  and  108|,  what  will  be 
the  face  of  a  bill  that  costs  $682.75? 

Answers:  $37.57,  $49.23,  $153.92,  $3253.15,  $536.50, 
fcs.  367.57,  fcs.  1217.27,  fcs.  2465,  fcs.  3221.82, 
fcs.  6650.82.'* $11.64. 


IMPORTING.  157 


XXX.    IMPORTING. 

Art.  1.  Importing  is  the  business  of  buying  goods 
in  a  foreign,  to  sell  in  the  home  market.  A  tax,  un- 
der the  name  of  Duties  or  Customs,  is  imposed  by  gov- 
ernment on  most  imported  articles  of  commerce. 

Such  taxes  are  levied  for  the  purpose  of  creating 
revenue  to  defray  the  expenses  of  government,  or  to 
protect  home  manufacturing  and  agricultural  inter- 
ests. Duties  are  regulated  by  a  scale  of  rates  called 
a  Tariff,  and  are  altered  according  to  the  exigencies 
of  the  times. 

A  high  tariff  signifies  high  rates  of  duties ;  a  low 
tariff,  low  rates  of  duties.  In  the  United  States,  a 
high  tariff  is  called  for,  when .  the  expenditure  of 
government  exceeds  the  revenue.  In  Great  Britain, 
it  is  advocated  when  imported  articles  sell  so  cheap 
as  to  interfere  with  the  sale  of  home  products. 

The  persons  appointed  to  examine  imported  goods 
and  collect  duties,  are  called  Custom  House  OfficerSy 
and  their  place  of  business,  the  Custom,  House. 

Art.  2.  Duties  are  of  two  kinds :  ad  valorem  and 
specific.  Ad  valorem  duties  consist  of  a  rate  per  cent, 
on  the  value  of  the  goods,  as  stated  in  the  invoice; 
specific  duties,  of  a  stated  sum  of  money  on  the 
quantity  imported,  without  regard  to  value,  as  $1  a 
gallon,  $20  a  ton. 

Certain  allowances  are  made  on  goods  charged 
with  specific  duties.  These  are  draft,  tare,  leakage, 
and  breakage.  These  allowances  sometimes  consist 
of  a  percentage  of  the  weight  or  quantity,  and  some- 
times of  a  specified  deduction. 

Tare  is  an  allowance  made  for  the  weight  of  the 


158  IMPORTING. 


box,  barrel,  bag,  crate,  etc.,  which  contains  the  goods, 
and  is  usually  calculated  by  percentage,  etc.,  after  the 
deduction  for  draft  is  made. 

Draft  or  tret  is  an  allowance  made  for  loss  by 
weighing  in  small  quantities,  and  for  impurities  to 
which  some  goods  are  subject. 

On  112  lbs.,  or  less,  it  is  1  lb.;  from  112  lbs.  to  224 
lbs.,  2  lbs.;  from  224  lbs.  to  336  lbs.,  3  lbs. ;  from  336 
lbs.  to  1120  lbs.,  4  lbs. ;  from  1120  lbs.  to  2016  lbs.,  7 
lbs. ;  more  than  2016  lbs.,  9  lbs. 

Note. — The  draft,  though  not  stated  in  the  question,  is  to  be 
deducted  before  other  allowances  are  made. 

Leakage  is  an  allowance  of  2  ^  on  liquids,  in  casks, 
paying  duties  by  Ihe  gallon. 

Breakage  is  an  allowance  on  bottled  liquors,  usually 
5  ^,  but  on  ale,  beer,  and  porter,  10  ^. 

Gross  Weight  is  the  total  weight  of  goods  and  box, 
barrel,  etc. 

Net  Weight  is  what  remains  after  all  deductions 
are  made. 

Art.  3.  Goods  imported  may  be  placed  in  Govern- 
ment warehouses  and  the  duty  paid  on  withdrawal 
therefrom,  or  duties  may  be  paid  at  once  and  the 
goods  taken  by  the  importer. 

2.  Entries  should  be  made  within  twenty -four  hours 
of  the  arrival  of  the  goods. 

3.  Each  class  of  merchandise  should  be  entered  by 
itself,  without  regard  to  rate  of  duty,  and  goods  de- 
signed for  warehouse  entered  by  the  case. 

4.  All  charges  incurred  before  shipment  at  the  for- 
eign port  should  be  added  to  the  invoice  value ;  and 
every  invoice  *  is  subject  to  duty  on  at   least   2\  ^ 


IMPORTING.  159 


commission.     Except  the  fee  for  Consul's  Certificate^ 
ocean  insurance,  and  freight,  which  are  not  dutiable. 

5.  Fractions  are  omitted  in  reckoning  duties.  Half 
a  dollar  is  considered  $1;  under  that  the  fraction  is 
rejected. 

6.  On  the  back  of  the  blank  form  of  entry  is  an 
affidavit  to  be  made  by  the  owner  or  owners  of  the 
merchandise,  or  some  one  acting  for  them  under 
power  of  attorney,  stating  that  the  invoice  produced 
is  the  only  invoice  received  for  the  goods ;  that  the 
entry  contains  a  true  account  of  said  goods;  that 
nothing  has  been  concealed  whereby  the  United 
States  may  be  defrauded ;  that  if  any  mistake  is  dis- 
covered in  the  future  the  affiant  will  make  it  known 
to  the  Surveyor  of  Customs,  etc. 

7.  There  are  three  separate  entries  on  the  blank — 
next  page — which,  in  practice,  would  be  made  out 
on  separate  papers. 

8.  The  first  entry  represents  merchandise  subject  to 
both  specific  and  ad  valorem  duties,  and  a  reduction 
of  10  %  on  the  rates. 

Art.  4.     Custom  House  values  of  foreign  currencies. 

Crown  of  Sweden,  Norway  i  Pagoda  of  Madras 1.84 

and  Denmark 268    |  Patacans  of  Uruguay 

Dollar,  Egypt 1.0039    '^  ""  '^   ' 

Dollar,  Mexican 1.0475 

Dollar,  Central  America..  .965 

Florin  of  Austria 476 

Florin,  Southern  Germany. 
Florin  of  Netherlands.....  .405 

Franc,  France  k  Belgium.  .193 

Lira.  Italv 193 

Mahbul),  Tripoli 8909 

Mark  reichs(rix),Ger.Em.  .2382 

Milreis  of  Brazil 5456 

Milreisof  Portugal 1.0847 


Peso  of  Cuba 9258 

Peso  of  Chili 9123 

Peso  of  Venezuela 7773 

Peso  of  Columbia : 965 

Pound  Sterling,  Gr.  Brit.. 4.8665 

Piastre,  Turkish 0439 

Rix  mark,  Germany 2382 

Rupee  of  India 4584 

Ruble  of  Russia 7717 

Tale  of  China 1.61 

Thaler  (see  dollar) 

Yan  of  Japan 997 


160 


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IMPORTING.  161 


Art.  5.     FOKEIGN  INVOICES. 
1.  Invoice  of  five  cases  woolens,  shipped  on  steamer 
<' Ocean,"  London,  November  18,  1876,  to  John  Jones 
k  Co.,  New  York. 

y     Five  cases  woolens,  weighing  500  lbs. 


^  /S.  £     s.  d.        £      s.  d, 

8471  yds.  @  16d 56  10  0 

819      ''      "   14d 47  15  6 

740      "      "  15d 46     5  0       150  10  6 

Shipping  charges,  .....  120 

Cases, 7  6          19  6 

Commission  @  2^  ^,  .     ...  3  16  0 

{amount  chargeable  with  duty,)  £155  16  0 

Consular  Certificate,  ....  0  10  0 
Ocean  Insurance  on  £150@328.%,  2  8  0 
Freight,  etc., 4  10  0  7     8  0 

£163  "4~b 

a.  What  is  the  duty  @  50c.  per  pound  less  10  %, 
specific,  and  @  35  ^  less  10  %  ad  valorem  duty? 

b.  What  is  the  entire  cost,  in  currency,  of  the  in- 
voice, gold  quoted  @  1.02  and  exchange  @  4.90? 

Note.— On  receipt  of  an  invoice  the  accountant  should  examine  every 
item  and  ascertain  whether  or  not  it  is  correct.  The  first  item  of  the  in- 
voice is  8473^  yds.  @  16d.=  13560  pence  or  £56  10;  the  second  item  is'819 
yds.  @  14d.=  11466  pence  or  £47  15  6;  the  third  item  is  740  yds.  @  15d.= 
11100  pence  or  £46  5. 

The  next  item  for  examination  is  the  ocean  insurance,  £150@32s.  per 
cent.  (32  shilling  per  cent,  is  32  shillings  on  every  £100;  i.  e.,  if  we  mul- 
tiply the  shilling  per  cent,  on  one  per  cent,  of  the  £  the  answer  will  be 
in  shillings),  one  per  cent,  of  £150  is  £1  50-100.  1.50X32  =  48;  i.  e.,  48 
shilling,  or  £2  8s. 

We  now  proceed  to  ascertain  the  duty.  The  ad  valorem  duty  on  the 
first  three  extensions  of  the  invoice,*  aggregating  £155  16  0,  which  mul- 
tiplied on  4.8665  the  custom-house  value  of  a  £)  =  $758.  @  35  per  cent  = 
$265.30,  less  10  per  cent.  =  $238.77.  The  specific  duty  on  500  lbs.  @  50  cents 
=?  ^250,  less  10  per  cent.  =  8225.    S238.77  +  S225  =  $463.77,  duty  in  gold. 

In  order  to  ascertain  the  entire  cost,  we  must  reduce  the  amount  of 
the  invoice  to  Federal  money,  and  add  the  dnty.  £163  4  0  reduced  to 
Federal  money  (according  to  Art.  2,  p.  152),  S4.90  being  the  value  of  a  £, 
would  be  $799.68  in  gold,  plus  the  duty,  $463.77  =  $1263  45.  Gold  being 
worth  $1.02  in  currency,  the  entire  cost  in  currency  is  $1288.72. 

*  See  Art.  3,  If  4,  page  156. 


162  IMPORTING. 

2.  Invoice  of  queensware  consigned  to  Mr.  W.  H. 
Hall,  Cincinnati,  O.,  by  W.  Anderson,  Liverpool, 
shipped  on  steamer  "Star,"  March  16,  1876. 

W\[c]  J  40  72  doz  Twiflers,  J  soups,  blue 

edge, Vs  6     0  0 

141  100  doz.  Muffins,  Tin.  flat, 

blue  edge, Ve  .    7  10  0 

142  One  crate  same, 7  10  0 

143  60  doz.  unh.    London   teas, 

neatly  painted, ^/^  6  15  0 

144  One  crate  same,  only  ajiother 

pattern, 6  15  0 

145  One   crate  same,  only  blue 

sponged  marble, 6   15  0 

146  One  crate  same, 6  15  0 

147  One  crate  same,  only   pink 

sponged  marble, 6  15  0 

•  16  8 

148  24  doz.  jugs  6^6,  12V-,  cone 

blue  dipt., 4     80 

40  doz.  unhd.  toy  teas,  fancy 

sponged, f /g  2  10  0       6  18  0 

149  38  doz.  bowls,  6,  9,  24,  30,  36, 

bluedipt,. Vs  6     3  6 

£ 

Packages 1  10  6 

Commission,  2  % 

Freight  and  Shipping  Charges...  2  12  0 

Ocean  Ins.  @  41s.  %  on  £70 

What  is  the  duty  @  45  less  10  ^,  including  com.  @ 
2 J  %,  and  what  is  the  entire  cost  of  the  invoice  in 
currency,  gold  1.02,  exchange  4.87-|? 

*  16  doz.  sixes  at  3  shillings  and  6  pence. 


IMPORTING.  163 


3.  Memorandum  of  goods  shipped  on  steamer  "Glad- 
lola,"  from  Liverpool,  March  18,  1876,  and  consigned 
to  E.  H.  Langdale,  Cincinnati,  Ohio. 

154  38  doz.  bowls,  6,  9,  24,  30,  36, 

C.  C, %  5    4  6 

155  24  doz.  jugs,  12,  63/-,  12V6) 

coneC.  C, 3  15  0 

40  doz.  unhd.  toy  teas,  fancy 

sponged, Vs  2  10  0      6    5  0 

156  60doz.  unh.Londonteas,C.C.,  Vio  5  10  0 

157  60  "      "  "         "      "  5  10  0 

158  One  crate  same, ,  5  10  0 

8  10 

159  50  doz.  nappies,  5%,   6V6» 

hu  '1%,    93/6,     10%, 

beaded  C.  C, 5  18  8 

160  72  doz.  twiflers,  flat  French, 

'     C.  C, 1/7  5  14  0 

161  100  doz.  muffins,    7  in.   flat, 

French  C.  C, Vs  7     1  8 

162  One  crate  same, 7     18 


Discount    5  ^  * 

. ._-Jto£l 

I  Gold  110. 


/  Exch  $485  to  £  1 


Packages,  3  16  0 

Charges,  8     7  0 

Cash,  March  31,  1876, 
What  is  the  duty  @  35  less  10  ^,  including  com., 
and  what  is  the  entire  cost  in  currency  of  the  invoice  ? 

♦The  pupU  wiU  fill  the  blanks. 


164 


IMPORTING. 


4.  Invoice  of  six  hogsheads  of  tobacco,  shipped  on 
board  the  "  Leviathan/'  Davis,  master,  and  consigned 
to  Edwin  Kessinger,  on  his  account  and  risk. 


No. 

fcwt. 

qr. 

lbs. 

cwt, 

.  qr. 

lbs. 

K.  K. 

1 

18 

0 

23 

2 

11 

I  to  6 

2 

19 

1 

12 

3 

5 

3 

18 

3 

15 

2 

26 

4 

18 

1 

26 

2 

19 

5 

19 

2 

24 

3 

24 

6 

12 

2 

17 

2 

17 

* 

* 

96     3     15  net,  @  7d.  per  lb. 

Charges. 

Bond  and  custom  house  entry, £0  10  6 

Cost  of  empty  hogsheads,  4  16  0 

Lighterage  and  small  charges,  1     4  0 

Bills  of  lading,  6  6 

Brokerage  £316  99@^%,     *  888 


Com.  on  £324  18  5  @  2  %,       * 
Insurance  on  £350  @  428.  %, 
Com.  on  same  ®  ^  %, 
Policy  duty,  1 


1  0 


£341  11  5 
Exch.  $5  currency  to  the  £; 
Errors  excepted 

C.  H.  Guiou 
Liverpool  J  Aug,  3,  1876. 

What  is  the  duty  @  40  less  10%  on  this  invoice? 

Answers:  Duty  on  the  several  invoices,  S140.13, 
$99.23,  $463.77,  S583.56.  Cost  in  currency,  $514.73, 
«446.52,  $2320.56,  $1288.72. 

«  The  pupil  will  fill  the  blanks.       f  Avoirdupois  weight. 


FARMING.  165 


XXXI.    FARMING. 

Art.  1.  The  business  of  a  farmer  sometimes  com- 
prises several  of  the  mechanic  arts,  in  which  a 
knowledge  of  arithmetic  is  necessary.  It  also  some- 
times embraces  surveying,  engineering,  etc.,  pro- 
fessions that  require  a  familiar  knowledge  of  this 
science. 

Art.  2.  To  find  the  number  of  acres  in  a  field 
or  tract  of  land  having  4  square  corners,*  we  multi- 
ply the  length  by  the  breadth,  and  divide  the  result 
by  160,  if  the  measure  was  taken  in  rods ;  or  by 
43560,  if  taken  in  feet.f 

1.  The  length  of  a  field  is  125  rods,  and  its  breadtli 
112  rods,  how  many  acres  are  in  it? 

2.  A  lot  of  land  is  400  ft.  long  by  110  ft.  broad, 
how  many  acres  does  it  contain  ?  * 

(For  answers  see  end  of  chapter.) 

Art.    3.     To  lay  off  a  given  quantity  of  land. 

What  should  be  the  length  of  a  strip  of  land  30 
rods  broad,  to  contain  6  acres? 

In  6  acres  there  are  960  rods,  which,  divided  by 
30=32  rods. 

Art.  4.  To  find  the  contents  of  a  triangular  field 
having  a  square  corner ,  (a  right-angled  triangle,)  we 
multiply  the  two  shorter  sides  together,  and  take 
one-half  the  product. 

Reason. — A  right-angled  triangle  is  half  a  square  or  paral- 
lelogram, formed  by  drawing  a  line  between  opposite  corners. 

t  In  a  sq.  rod  there  are  272J-  sq.  feet.  When  there  are  feet  re- 
maining to  be  reduced  to  rods,  it  will  be  sufficiently  accurate  to 
divide  by  272. 

**  A  figure  having  square  corners,  and  all  its  sides  equal,  is  a 
tquare ;  one  having  its  opposite  sides  equal,  a  rectangle  or  paral- 
lelogram. 


166  FARMING. 


1.  The  shorter  sides  of  a  right-angled  triangle  are 
45  and  60;  required  the  contents. 

Art.  5.  To  find  the  quantity  of  grain  or  coal  in 
a  bin  or  wagon,  we  multiply  the  length,  breadth  and 
hiirht  together;  and  for  grainy  divide  the  product  by 
1.2444,*  if  the  dimensions  were  given  in  feet;  or  by 
2150. 42. t  if  o;iven  in  inches.  For  coal,  by  1.555  or 
2688. 

To  find  the  number  of  bushels  of  unshelled  corn  in 
a  bin,  we  multiply  the  cubic  feet  by  4^  and  divide  the 
product  by  10. 

1.  A  wagon  is  8  feet  long,  5  feet  broad,  and  18  in. 
deep,  how  many  bushels  of  corn  does  it  contain  ? 

8  X  5  X  li^^^^^j  ^^^^  number  of  cubic  feet, 
60  :  1.2444  =  48.21  or  48^  bushels. 

2.  How  many  bushels  of  grain  in  a  bin  measur- 
ing 4  feet  every  way? 

Art.  6.  To  find  the  quantity  of  wood  or  bark  in 
a  pile,  we  multiply  the  three  sides  given  in  feet 
as  before,  and  divide  by  128,  the  number  of  feet  in 
a  cord. 

1.  How  many  cords  of  wood  in  a  pile  40  feet  long, 
7  feet  high  and  4  feet  broad? 

Art.  7.  Having  two  sides  and  the  contents  of  a 
box,  to  find  the  third  side,  we  divide  the  cubical  con- 
tents by  the  product  of  the  two  sides. 

Reason. — Since  the  product  of  the  three  sides  equals  the  con- 
tents, then  the  contents  divided  by  two  of  the  sides  will  give 
the  third  side. 

*  Feet  in  a  bushel  t  Inches  in  a  bushel. 


FARMING.  167 

1.  A  box  is  2  feet  wide  and  3  feet  high,  how  long 
should  it  be  to  hold  25  bushels  of  coal? 

In  -25  bushels  there  are  2688X25  or  67200  cu.  in. 
In     2  feet  there  are  24  inches. 

•  •      ;]     ^^       ^'        ^'     36        "        24X36  =  864=  area 

[of  the  end 

864)67200(77  inches,  or  6  feet  5f  in.  length  of  box. 
6048 


6720 
6048 

672 
554 


2.  What  must  be  the  hight  ot  a  bin  that  will  hold 
300  bushels  of  wheat,  if  its  length  is  30  feet,  and  its 
width  4  feet? 

3.  What  must  be  the  depth  of  a  box  16  inches 
square  to  hold  a  bushel;  a  box  10  inches  square  to 
hold  a  peck  ;  one  8  inches  square  to  hold  half  a  peck  ? 

To  find  the  side  of  a  cube  that  will  hold  a  certain 
quantity. 

Art.    8.      To  find  the  quantity  of  grain  when  heaped 
against  a  wall  or  partition.     Take    half  the  perpen- 
•dicular  hight  for  one   side  and  multiply  it  by  the 
length  and  breadth,  as  in  Art.  5. 

Art.  9.  To  find  the  number  of  cubic  feet  in  a 
round  log. 

How  many  feet  are  in  a  log  12  feet  long  and  30 
inches  diameter? 

In  30  inches  there  are  2J,  or  2.5  feet:  2.5X2.5X 

,7854=4.9087,    the   area    of  the  end.     4.9087X12  = 
58.9044,  or'58/jj  feet,  the  solid  content. 


168  FARMING. 


Note. — This  method  of  calculating,  though  correct,  is  seldom 
used  for  practical  purposes.  It  is  customary  for  lumber  merchants 
to  throw  off  J  of  the  diameter,  and  consider  the  remainder  the 
side  of  a  square  log.  A  log  of  the  dimensions  named  in  the  pre- 
ceding question,  would  thus  measure  only  33J  feet,  or  J  of  100 
feet;  and  is  thereby  taken  as  the  standard  of  measurement  in 
some  of  the  Western  States.     See  Lumber  Business  page  169. 

Art.  10.     Trade  or  Barter. 

1.  How  many  cords  of  wood  at  $3.75  a  cord  should 
I  get  for  50  bushels  of  wheat  at  $1.12^  a  bushel 

50X1. 121=856. 25,  which  divided  by  S3. 75,  will 
give  the  number  of  cords. 

5625  :  375  =  15  cords.— Proof— 15  cords  @  $8.75  = 
$56.25. 

2.  How  many  pounds  of  sugar  at  8  cents  a  pound, 
should  I  get  for  127  lbs  of  butter,  at  121  cents  a 
pound? 

3.  How  many  days  work  of  a  man  at  75  cents  a 
day,   will  be   equal  to  45   days   work   of  a   man   at 

$1.25? 

4.  How  many  cords  of  wood  at  $2.25,  will  be  equal 
to  150  cords  at  $3.50? 

To  Determine  the  Weight  of  Live  Cattle. — 
Measure  in  inches  girth  around  breast  just  behind 
shoulder  blade,  and  the  length  of  back  from  tail  to 
fore  part  of  shoulder  blade.  Multiply  girth  by  length 
and  divide  by  144.  If  girth  is  less  than  three  feet, 
multiply  quotient  by  11;  if  between  three  and  five,  by 
16 ;  between  five  and  seven,  by  23 ;  between  seven  and 
nine,  by  81.  If  animal  is  lean,  deduct  one-twentieth 
from  result ;  or,  take  girth  and  length  in  feet,  multi- 
ply square  of  girth  by  length,  and  multiply  product 
by  8.36.  Live  weight  multiplied  by  .005  gives  net 
weight — nearly. 

Answers:  15,  198J,  75,  31    87J,  51^  77,  8f,  1350, 
•    1  ac,  1.7  rds.,  48i,  350,  93|,  87^ 


LUMBER  MEASURE.  169 


XXXII.    LUMBER  MEASURE. 

Art.  1.  Lumber  measure  comprises  solid  and 
superficial  measure.  Hound  logs  are  measured  by 
deducting  one-third  of  the  diameter  for  waste,  and 
calling  the  remainder  the  side  of  a  square  log. 

To  find  the  contents  of  a  round  log  24  inches 
in  diameter,  and  30  feet  in  length. 

Solution. — Deducting  J  from  24  for  waste,  we  have  16,  which 
squared  =256in.  and  multiplied  by  the  length  =640  ft.  hoard 
measure. 

In  some  places,  only  \  is  deducted  for  pine  lumber. 

Note. — Inch  measure  is  taken  as  the  standard  for  lumber.  If 
a  board  is  under  an  inch,  it  is  measured  as  a  full  inch ;  and  if  over 
an  inch,  it  is  reduced  to  inch  measurement.  A  plank  2  inches 
thick,  would  be  considered  as  2  boards  one  inch  thick. 

Planks  or  joists  are  sometimes  reckoned  by  face 
measure,  that  is,  the  dimensions  of  one  side  of  the 
board  are  taken  instead  of  the  solid  content.  A  16 
feet  board  2  inches  thick  by  12  inches  broad,  would 
measure  32  feet,  hoard  measure,  or  16  feet /ace  measure. 

In  some  places,  the  saw  log  is  taken  as  a  standard 
of  measurement  for  round  timber.  A  log  12  feet 
long  and  30  inches  in  diameter,  is  the  standard  in 
some  parts  of  the  west.  In  Pennsylvania,  a  saw  log 
is  one  that  will  cut  into  200  feet  of  lumber. 

Boards  of  different  widths  are  measured  with  a 
tape-line,  as  they  are  put  on  the  wagon,  by  passing 
one  hand  to  the  other  in  measuring  each  board.  If 
the  boards  are  12  ft.  long,  the  number  of  inches  meas- 
ured will  be  the  number  of  feet  of  lumber ;  in  measur- 
ing 16  ft.  boards,  one-third  must  be  added. 


170  LUMBER   MEASURE. 


xiRT.  2.  To  measure  timber  partly  squared,  ot 
having  its  ends  of  the  form  of  the 
diagram,  it  is  customary  to  deduct 
the  *'wane"  (the  length  of  the 
corner),  from  the  thickness  of  the 
log,  and  call  the  remainder  one 
side.  A  log  18  inches  thick,  with 
a  "  wane "  3  inches,  would  be 
called  one  of  18  by  15  inches. 


18  inches. 


1.  In  an  octagonal  log  25  feet  long,  20  inches 
thick,  with  a  wane  4  inches,  how  many  solid  feet 
are  there? 

2.  There  are  150  logs,  the  average  length  and 
breadth  of  which,  are  20  feet  by  22  inches,  wane  3 
inches  ;  required  the  number  of  solid  feet  they  contain. 

3.  In  a  raft,  there  are  450  boards  16  feet  long  and 
i^  inches  thick,  and  measuring  in  the  aggregate  757 
feet  broad  ;  how  many  feet  of  lumber  (board  measure) 
does  it  contain?    How  many /ace  measure? 

4.  How  much  lumber  can  be  cut  from  a  tree 
measuring  20  feet  long  and  14  inches  diameter  at 
the  smaller  end,  allowing  for  waste,  one-fourth  of 
the  diameter? 

5.  The  average  length  of  50  logs  is  21  feet,  and 
the  average  thickness  24  inches,  wane  2  inches; 
required  the  number  of  solid  feet  they  contain. 

Answers:  8708J,  55|,  18.168,  12.112. 


FRACTIONS.  171 


XXXIII.    FRACTIONS. 

Art.  1.  When  numbers  are  written  as  ibll(n\s. 
they  are  called  fractions:  ^,  |,  .05. 

A  common  fraction  is  composed  of  two  terms  or 
numbers,  one  above  the  other,  with  a  line  between 
them,  as  ^,  f ;  which  signify  one  part  of  something 
which  has  been  divided  into  two  parts,  and  three  parts 
of  something  which  has  been  divided  into  four  parts. 

The  upper  term  is  called  the  numerator,  the  under 
one,  the  denominator. 

h  T3J  //o'  ilh-Qy  would  be  read  one-half,  two-thir- 
teenths, sixty-seveyi  five  hundred  and  twentieths,  eleven 
two  thousandths. 

Art.  2.  Common  fractions  are  divided  into  simple, 
proper,  improper,  compound,  complex,  fractions,  and 
mixed  numbers. 

A  simple  fraction  is  a  single  fraction,  as  f,  \^, 

A  proper  fraction  is  a  single  fraction  whose  numer- 
ator is  less  than  the  denominator,  as  ^. 

An  improper  fraction  is  a  single  fraction  whose  nu- 
merator is  equal  to  or  greater  than  the  denominator, 
as  I,  §• 

A  compound  fraction  is  a  fraction  -of  a  fraction  or 
fractions,  as  J  of  f ,  |  of  ji  of  |§. 

A  complex  fraction  is  one  having  a  fraction  in  th(? 

numerator  or  denominator,  or  in  boti),  a^  -,    — . 

7'    i 

A  mixed  number  is  composed  of  a  fraction  and 
whole  number  together,  as  7f. 


172  REDUCTION    OF   FRACTIONS. 


Art.  3.  REDUCTION  OP  FRACTIONS. 

Reduction  of  fractions  consists  in  changing  the 
terms  without  altering  their  values;  thus,  f^  can  be 
reduced  to  |,  and  that  to  i,  without  altering  the 
value  of  the  fraction. 

To  reduce  a  fraction  to  a  lower  denomination,  we 
divide  the  two  terms  by  such  a  number  as  will  do 
so,  without  a  remainder. 

1.  Reduce  f  |  to  its  lowest  terms. 

mKi\  and  3)T^.(i 

Explanation. — 24  and  96  were  divided  by  8,  giving  the  quo- 
tients 3  and  12.  Then  3  and  12  were  divided  by  3,  giying  1  and 
4,  or  J.     See  Prop,  of  Numbers,  page  82. 

2.  Eeduce  f-^-^-^  to  its  lowest  terms. 

5)Tm=5HM=5)||=f 

3.  Eeduce  the  following  fractions  to  their  lowest 
terms : 

24      184       2)        J  25         6874         7216         234         126       2168 
5155   T3B>  1¥5>   tTJCiO'  lZ4gH5»    Sg^HOJ    5155'    5555?   ?5S3?> 
9  3  6  9  8  7  6 

Answers  arranged  promiscuously  ^,  J,  jfyy,  iHfj 

g       3,     JL     2  3      _4  9  1         1         9 
5>   25J    8)    ^l>    llQZ)    ■&)  3J38 

Art.  4.  To  reduce  a  mixed  or  whole  number  to  an 
improper  fraction.  This  is  done  in  the  same  way  that 
we  would  reduce  feet  and  inches  to  inches. 

1.  Eeduce  5 J  to  an  improper  fraction;  that  iSj  in 
5-|  how  many  eighths? 

5X  Explanation. — In    every  whole  number    there  are   8 

g  eighths,  and  in  5  whole  numbers  there  are   8  times  6,  or 

40  eighths,  to  which  add  7   eighths,  and  the  result  is  4? 

47  eighths. 

Eeduce  the  following  numbers  to  improper  frac- 
..ions:  7|,6|,  5l|,  ITg^,  HS^g,  16^. 

Answers.— ^,^,  V,  1*,  'W,  ^V",  Vc'- 


FRACTIONS.  178 


Art.  5.  To  reduce  improper  fractions  to  whole  or 
mixed  numbers^  is  an  operation  the  reverse  of  the 
last. 

1.  Eeduce  y  to  a  mixed  number. 

47  —  PSX 

2.  Eeduce  the  following  to  whole  or  mixed  num- 
bers, and  the  remaining  fractions  to  their  lowesi 
terms : 

146  456   364  161  J196   100  4160  3179  7854   llOOO 
g  >   5  »   5  >  15  J  21»  >   5  >   g   »  1^'E  J  H64  »    S6 

Note. — When  fractions  are  to  be  reduced  to  their  lowest 
term,  and  the  learner  should  be  unable  to  see  what  number 
both  the  denominator  and  the  numerator  can  be  divided  by,  the 
Greatest  Common  Divisor  may  be  found  by  dividing  the 
denominator  by  the  numerator,  and  the  numerator  by  the  re- 
mainder, and  the  old  remainder  by  the  new  remainder,  etc. 
The  last  divisor — i.  e.,  the  one  which  can  be  divided  into  the 
dividend  without  a  remainder  is  the  Greatest  Common  Divisor, 


130)169(1  To  find  the  greatest  common  divisor  of 

130  -Jf^,  we  proceed  as  follows: 

39)130(8  13  being  the  divisor,  that  divided   39 

117  without  a  remainder,  is  the  greatest  com- 

~TF')39C3  ^^on  divisor:    130-^-13  =  10 

39  'l69--13  =  13 
MULTIPLICATION  OF  DECIMALS. 

Art.  6.  In  this  rule  we  multiply  as  in  whole 
numbers,  and  mark  off  as  many  places  of  decimals 
in  the  product  as  there  are  in  the  two  factors. 

1.  5.7X6.107 

6.107     There  are  3  places  in  this  factor, 
5.7  and  1  place     in  this       " 

42749 
30535 


34.8099     so  we  mark  off  4  in  the  product. 
Answers:     182,   10i|,  24J,  50f,   12^    693J,  17^3^, 
129^,  9^,  47|i. 


174  DIVISION    OF    DECIMALS. 

Art.  7.  When  the  product  contains  fewer  figures 
than  there  are  decimals  in  the  factors,  we  make  up  the 
number  by  annexing  ciphers  to  the  left. 

2.  100X.0005. 

^QQ^'  The  product  contains  only  3  figures 

1 (^00),  so  we  annex  one  more  cipher  to 

500         make  up  four,  the  number  contained 
or  .0500        in  the  factors. 
Arts.  .05 

.107X.05     =* 
61.04X.0007= 
.7103X.004  = 


Total,  .0509192 

DIVISION  OF  DECIMALS. 

Art.  8.  Division  of  decimals  is  effected  in  the 
same  manner  as  division  of  whole  numbers,  with 
the  diflTerence  in  using  the  decimal  point.  77ie  divisor 
must  contain  as  many  'places  of  decimals  as  the  dividend, 

1.  To  divide  34.8099  by  6.107 

6.1070)34.8099(5.7  The  dividend  contains  4  de- 

30  5350  cimals,  and  the  divisor  only  3; 

— 7:7,770  so  we  point  off  one  in  the  quo- 

_flL_  in  Multiplication. 

Note. — When  a  remainder  occurs,  we  may  annex  ciphers  in- 
definitely, and  carry  out  the  quotient  to  as  many  places  as  we 
desire. 

Art.  9.  When  the  dividend  does  not  contain  as 
many  decimals  as  the  divisor,  annex  ciphers  to  the 
right  of  the  former,  until  it  contains  the  same  num- 
ber;  the  quotient  will  then  appear  in  whole  num- 
bers.  If  a  remainder  occurs,  annex  ciphers,  and 
the  result  will  be  decimals. 


FRACTIONS. 


175 


2.  Divide  3066  by  .1783. 

.1783)3066.0000(17195.7 
1783 


12830 
12481 

~^90 
1783 

17070 
16047 

1l0230 

8915 

13150 
12481 


Four  ciphers  have  been  an- 
nexed to  the  dividend,  and  a 
fifth  annexed  in  finding  the  7 
of  the  quotient;  so  we  point  off 
1  decimal. 


When  there  are  not  figures  enough  in  the  quotient 
to  make  up  the  number  of  decimals  in  the  dividend, 
annex  ciphers  to  the  left  of  the  former. 

3.  I>ivide  10.70067  by  370.4. 


370  0^0.70067(.0288 
7408 


32926 
29632 

"32947 
29632 


Here  the  quotient  pro- 
duced only  three  figures 
(288),  which,  with  the  one 
in  the  divisor  makes  only 
four  decimals;  so  to  make 
the  number  equal  to  the 
decimals  in  the  dividend 
we  annex  a  cipher  to  the  left 


314.06-~10.73  =29.2693 
17600-^785.4  = 
3170.09-^2.4014= 
417.456-^31.145  = 


Total,  1385.1825 


176  REDUCTION  OF  DECIMALS. 

30.f)'40~-493.67  = 

10.8739^117.406  = 

6.342-^-22.973  = 

1467.06-j-196.04  = 


Total,  7.91420 

REDUCTION  OF  DECIMALS. 
Art.  10.    To  reduce  a  common  fraction  to  a  decimal^ 
we  annex  ciphers  to  the  right  of  the  numerator,  and 
proceed  as  in  division. 

Reduce  ^  to  a  decimal,  f  to  a  decimal. 
2)10  4)300 

75  775 

Eeduce  ^  to  a  decimal. 
3)100000 

.3333         This  quotient  might  be  carried  out  in 
definitely.     It  is  called  a  repeating  decimal.     Such  h 
decimal  is  marked  thus,  .3.     Its  fractional  value  is 
restored  by  using  9  instead  of  10  for  the  denomina- 
tor.     3  =  1. 
Reduce  4  to  a  decimal. 

7)1000000000000  This  is  called  a  circulating 

142857142857         decimal,  and  is  marked  thus, 

i  45?85'7 1  4  -2  8  5  7  __  1 

Its  fractional  vulue  is  restored  in  the  same  manner 
as  that  of  the  .3  in  the  preceding  example. 
Express  the  following  fractions  decimally  : 

33  1 

%  i 

1  6  1 

5  4  3 '71 
1  4 

3  3  T7 


Total,  1.1186  Total,  1.2239 


FRACTIONS.  177 


Art    11.     To  find  the  value  of  the  decimal  part  of  a 
denominate  number^  as  £0.75,  $0.33 J,  etc. 

1.  What  is  the  value  of  .5  of  a  yard? 

If  we  wanted  to  know  the  value  of  5  yards  in  a 
lower  denomination,  we  would  multiply  by  4,  as  5X 
4=20;  that  is  20  quarters.     The  same  principle  ap 
plies  in  decimals. 

.5 
4 

2.0  Ans.  2  quarters. 
£0.345  is  how  much? 

,  345  Explanation. — The  next  lower  denom- 

2Q  ination    to    pounds    is  shillings;    so   we 

multiply  by  20,  and  point  off  three  figures 

6.900  shillings,  ^o   correspond   with    the   number   in    the 

22  factors.     The  next  lower  denomination  is 

, pence;  so  we  multiply  by  12,  and  the  next 

10.800  pence.  farthings,  which  we  multiply  by  4.     The 

^  answer  is  6  shillings,  10  pence  3^  farthings, 

3.200  farthings 

Note. —  1.  The  ciphers  on  the  right  need  not  be  used,  as  they 
possess  no  value. 

2.  Observe  that  the  whole  numbers  are  not  multiplied,  else  the 
shillings  and  pence  would  be  reduced  to  farthings. 

Find  the  value  of 

3.  .625     of  a  gallon.  Ans.    2,  1. 

4.  .1425  of  a  year.  Ans.     1,  21  3  . 

5.  .8323  of  a  £.  Ans.  16,  7|. 

6.  .1374  of  a  tun.  Ans.  274,12.8. 

7.  .0037  of  a  lb.  Troy.  Ans.        21.3. 

Art.  12.    To  reduce  denominate  values  to  decimals. 

Eeduce  6  shillings,  10  pence,  3i  farthings  to  the 
decimal  of  a  pound  sterling. 


178  DECIMALS. 


1.  5)    10 

4)'~372 
_ — 1_  This   operation  is  the  reverse  of  the 

12)10.8       last.     Observe  that  the  3  in  the  second 

2Q\   g  Q        line  was  annexed  after  reducing  i  to  a 

I 1_        decimal;  so  with  the  10  and  6. 

£  .345 

2.  Eeduce  3  quarters  to  the  decimal  of  a  yard. 

3.  Eeduce  6  lbs.  3  oz.  to  the  decimal  of  a  cwt. 

4.  Eeduce  12s.  6|d.  to  the  decimal  of  a  £. 

5.  Eeduce  12  lbs.  to  the  decimal  of  a  tun. 

6.  Eeduce  1  foot  3 J  in.  to  the  decimal  of  a  yard. 

7.  Eeduce  16  oz.  to  the  decimal  of  a  tun. 

The  pupil  can  prove  his  calculations  by  last  Art. 

PRACTICAL  QUESTIONS. 

1.  At  56  cents  a  pound,  what  will  127  lbs.  6  ounces 
of  tea  come  to  ? 

16)60 


.375  decimal  part  of  a  pound. 
127.375 
56 


764  250 
6368  75 


7133.000  or  $71.33 

Remark. — This  is  not  the  shortest  method  of  computing  the 
above ;  the  object  being  merely  to  show  the  practical  application 
of  decimals. 

2.  At  15  for  a  pound  sterling,  what  will  be  the 
value  of  £16  8s.  lOd.?  Arts.  $82.21. 

3.  What  will  be  the  value  of  the  following  sums 
9f  money,  at  the  same  rate? 


FRACTIONS.  179 


£167  10s.  3Jd.,  £19  2s.  6d.,  £10  10s.  lOJd. 


Total,$985.91 

4.  At  $75  per  hundred  (112  lb.),  what  will  14  cwt., 
3  qrs.  and  15  lbs.  cost? 

5.  Find  the   cost  of  3  tuns  15  cwt.  of  hemp,  a 
S140  per  tun  r' 

Answers.--U116.29j  $525. 


MULTIPLICATION  OF  COMMON  FEACTIONS. 
Art.   13.    A    fraction    is  multiplied    by   a   whole 
number,  by  simply  multiplying  the  numerator  with- 
out altering   the   denominator,     fx 7=7X3,   or   '\^j 
which  reduced  to  a  mixed  number,  equals  5|-. 

Art.  14.  Fractions  can  also  be  multiplied  by  di- 
viding the  denominator,  without  altering  the  numer- 
ator.    -4.X5=5)/3=|,  or  IJ. 

Multiply  the.  following  fractions: 

1.  |x   5=14         4.  /^X  11=2.357 

2.  |x   4=3|         5.  j\X    9=3.316 

3.  |xl2=8  6.  j%X   6=2.824 

Art.  15.  Mixed  numbers  are  multiplied  by  whole 
nutnbers,  as  compound  numbers  are  multiplied.  Let 
it  be  required  to  multiply  4|^  by  7. 

Whole  Nos.    Eighths. 
Illustration^  4  5  Explanation. — Seven   times  6 

tj  eighths    equal    35    eighths,  or    4 

^_  whole    numbers    and    3    eighths. 

32  3  Seven   times  4=28  and   4  make 

or    321  32.     Ans.S2l. 

°  It  will    not    be    necessary    for 

the  pupil  to"  write  his  work  in  so 
formal  a  manner  as  in  this  illua- 
tration. 


180  FKACTIOXS — ADDITION    AND    SUBTRACTION. 

2.  Multiply  61  by  12. 

12 

813 

Answers.  Answers. 

3.  6f     X    8=54  7.  914|   X  120=109760. 

4.  7'i     X   7  =  50|  8.     63^  X    15=       952.5 

5.  8|     X    6=53|  9.  127y\X   20=     2543.636 

6.  IJ/^X  12=13/^^3     10.  110^  X    14=     1542.333 
Art.  16.     To  multiply  fractions  together,  we  multi- 
ply the  numerators  together  for  a  new  numerator, 
^,nd  the  denominators  together  for  a  new  denomin- 
ator. 

1.  fXf=A,  orf. 

These  operations  might  have  been  abbreviated 
by  what  is  called  cancellation.  In  the  first  example, 
for  instance,  f  is  to  be  multiplied  by  |,  that  is,  the 
numerator  2,  is  to  be  multiplied  by  the  numerator  3; 
but  the  2  is  also  to  be  divided  by  3,  for  f  signifies  that 
2  is  to  be  divided  by  3 ;  therefore,  since  the  2  is  to  be 
multiplied  by  3,  and  divided  by  3,  it  remains  exactly 
the  same,  and  the  3  of  the  denominator  is  said  to 
cancel  or  make  void  the  3  of  the  numerator.  In  the 
following  operations,  the  canceled  figures  will  be 
known,  by  having  a  line  drawn  across  them. 

2     $     2  In  the  second  operation,  the  2  of 

tXz==~  the  numerator  and  the  6  of  the  de- 

nominator are  uncanceled,  making  |, 
2  $  ^  $  which  reduced  by  dividing  both  by  2, 
^^4^ 2^ "a     e^quals  J. 

^  .  The  2  and  6  might  have  been  can- 

^  $  4  $  celed  also,  by  dividing  both  where 
tX-X-X-    they  stood  in  the  question,  as  in  the 

^  o  3d  example,  placing  only  3  as  a  de- 
nominator,   and    1  as   a    numerator. 


FRACTIONS.  181 


1  is  alwa^^s  to  be  understood,  where  a  number  ha» 
been  canceled. 

1  Some   prefer  arranging  the  terms 

1$  ^  3     of  canceling   fractions    as   in    the 
15  margin,  with   the  denominator  or 

8     t^  37       divisor  on  the  left,  and  the  numer- 
ator on  the  right. 

Explanation. — The  first  2  was  canceled  in  the  18,  leaving  9; 
the  24  and  9  were  canceled  by  dividing  both  by  3,  leaving  8  and 
3;  the  74  was  canceled  by  the  second  2. 

The  fractions  arranged  in  the  usual  order  are 

Answers,  Answers, 

3.  4x  I  X  f  =i.  7.  l^X   2iXy«^=3.38 

4.  fXiXi|X/,=i  8.  6iX      |Xif=4.8 

5.  iXAX^,X^\=   .0041  9.  87iX^f.X^^=   .05 

6.  iXliX  f  X  i        .0265  10.  ^fiX52|x  4  =5.75 

Art.  17.  Compound  fractions  are  reduced  to  simple 
ones,  by  multiplication.  Let  it  be  required  to  reduce 
J  of  f  of  f  to  a  simple  fraction.  We  know  by  in- 
spection, that  one-half  of  f  is  J,  and  that  J  of  f  is  J, 
the  answer. 

By  multiplication  ^XfXf=3^^=J. 

I     $     $ 
By  cancellation     -X-X-=i. 

^  ^     $     4     ^ 

Answers.  Answers^ 

2.  I  of  I  of  ./y=J  6.  f  of  f  of  9|  =7.389 

3.  4  of  If  of  34=. 3  7.  fof  J  of /-J  =.0068 

4.  I  of  ^f  of  A  =  .286  8.  41  of  IJ  of  J   =1.5 

5.  4  of  1-1  of   f  =  1.75  9.     |of    fX    fXli=.4286 

10.  At  11}  cents  a  pound,  what  will  147^  lbs.  of 
coffee  cost? 

11.  What  will  7^  lbs.  of  cheese  cost,  at  9i  cents 
per  lb.  ? 


182  FRACTIONS — DIVISION . 


12.  At  12J  cents  a  pound,  what  will  120  lbs.  of 
sugar  cost  ? 

13.  What  will  14^  lbs.  of  beef  cost,  at  6|  cents  a 
pound  ? 

14.  Fifteen  and  a  half  yards  of  muslin  at  9J  cents, 
will  cost  how  much? 

Ansivers.— 71  cents,  $14.80,  $16.59,  98  cents,  $1 .43. 

DIVISION  OF  COMMON  FEACTIONS. 

Art.  18.  Division  being  the  reverse  of  multipli- 
cation, to  divide  a  fraction  by  a  whole  number ^  we  divide 
the  numerator  or  multiply  the  denominator. 

3)6               6 
<L-:_3=   1 =_2_     

^'  '  21      ''''   2lX3=gS=2T- 

Divide  if  by  4r^|     ||-4-  7=.1887 

i^  by  7=4     -if^l0=.076 

If  by  3=4     If-  4=.16 

Art.   19.     To  divide  mixed  numbers. 

1.  2lf-^6. 

"Whole  Nos.      Fifths. 

6)21             3  Explanation. — 6  is  contained  in    21    3 

times  and   3  left.     In   the  3  of  remainder, 

3             3  there   are  15  fifths,  which   added    to  the   3 

or  3-1  fifths  in  the  question,    make  18  fifths.     6  in 

^  18,  3   times.     Ans.    3|. 

2.  124-^8. 

8)124  Explanation. — In  this  example,  we  had  4  re- 

mainder,  in  which  were  28  sevenths,  and  the  one 

1||      in  the  question  made  29.     Then  as  8  would  not 
divide  29  without  a  jemainder,  we  multiplied  it 
n  the  denominator,  which  made  56.     Ans.  1^|. 

Answers.  Answers. 

3.  67^-T-  7=  9i|  7..  167^-^25 =6. 684 

4.  44|-^  3=14i|  8.     2l|~14=1.524 

5.  119|-^  6=19i|  9.     161-f-  7=2.306 


6. 


118|J-12=  9|^  lo!     22|4-12=1.'861 


FRACTIONS.  18B 


Art.  20.     One  fraction   is  divided  by  another,  by 
inverting  the  terms  of  the  divisor.     4-T-y=JX  J-=|-. 

Froof :  \-r-j  is  ^''divided  by  4,  because  f  is  4,  and 

by  Art.  we  find  that  -^-^-4=-       =4. 

'l  ^  2X4     ^ 

Caution. — The  pupil   will  observe   not  to    invert    the  terms  of 
the  number  to  be  divided. 

Complex  fractions   are  unsolved  questions  in  di- 
vision. 

— =2^-f-f=f  X|j  which  canceled 

7 

$    7 

2     $     ^  ^ 

1.  f-f=f  5.     |off--|=.53- 

2.  5^4=5|  6.  1JX*--|=.3 

.  3.  f-^^  =2.571     7.  2i  X  4~f  of  f=.6 
4.  |-f-|i=   .36       8.  3^     I 

^X?-^  of  ^=1.185 

9.  If  120 J  lbs.  of   cheese  cost  $14.80,  what  will  1 
lb.  cost? 

10.  Find  the  cost  of  1  lb.  of  coffee,  when  15Jlbs. 
cost  $1.43. 

11.  If  llj  yds.  of  cassimere  cost  $16.59,  what  will 
one  yard  cost? 

12.  If  9J  yds.  of  muslin  cost  71  cents,  what  will 
1  yard  cost?  Ansivers,—! ^%,  9/y,  147/^,  12^6/^ 

SUBTRACTION  OF  COMMON  FRACTIONS. 
Art.  21.     Fractions  and   mixed    numbers  can   be 
ubtracted  from   whole    nufnbers,  in   the  same  way 
hat  one  denominate  number  is  subtracted  from  an- 
ther. 


184  FRACTIONS — STJBTRACTION. 

From  87  take  25|. 

Whole  Nob.  Sevenths. 

Illustration.        87  0 

25  3 


61  4 

or  614. 

Explanation. — The  3  sevenths  could  not  be  taken  from  the 
number  above,  so  w^e  borrow  1  from  the  vy^hole  numbers,  in  which 
there  are  7  sevenths;  3  from  7  leaves  4.  Then  1  to  6  of  the 
whole  numbers  makes  6,  which  subtracted  from  7  leaves  1 ;  and  2 
from  8  leaves  6,  making  the  remainder  61^. 

Answers. 

2.  From  210  take  37^=172^ 

3.  '^      119     ''    821=36.875 

4.  ''        61     ''       4|=56.i 

5.  "        54     "       51=48.875 

6.  1063— 819»  243.8 

7.  3785— 1  Of  3774.14 

8.  2168— 14f  2153.571 

9.  1765—77711  987.4 

Art.  22.  To  subtract  one  fraction  from  another,  it 
ifl  necessary  that  both  be  of  the  same  denomination. 
We  can  subtract  \  from  f ,  but  can  not  conveniently 
subtract  J  from  f ,  without  first  altering  the  denomi- 
nation of  one  or  both  of  the  fractions.  Let  us  alter 
the  denomination,  and  reduce  both  fractions  to 
twelfths : 


1X4 

3X3      ^ 

4 

3X4     '^' 

4X3     '' 

Subtracting   these  four-twelfths  from  nine-twelfths, 
we  have  five-twelfths  for  a  remainder. 

Remark. — The  12  being  common  to  both  of  the  new  fractions, 
is  called  the  common  denominator. 


FRACTIONS.  185 


Art.  23.  To  find  a  common  denominator,  it  is  only 
necessary  to  multiply  the  denominators  all  together. 
The  common  denominator  of  -J-,  f,  f,  is  4X3X2=24, 
and  to  raise  fractions  to  a  common  denominator,  we 
simply  take  the  fractional  part  of  the  denominator, 
as  I  of  24=12,  that  is  ^|,  f  of  24  =  16,  that  is  ^|.  | 
€f  24=:18,  that  is  ^|.  The  fractions  reduced  to  a 
common  denominator,  are  ^|,  ^|,  ^|. 

1.  From  f  take  ^.      8X9=72  com.  denominator. 
I  of  72=63,  f  of  72=64.     f|— 11=^^5  Ans, 

Note. — A  shorter  method  of  finding  the  new  numerators  when 
there  are  only  two  terms,  is  to  multiply  the  first  numerator  on 
the  second  denominator,  and  the  second  numerator  on  the  first 
denominator. 

2.  From  5^  take  §. 

(41X5)— (8X2)     205—16 

41 2 : '       ^  ' 1  8  9  __42  9 

3.  ^f|-^ll  =  -,V5  8.  2i-l^^,  =  1.4 

4.  I  -t\=H  9.  ioff-i=.25 

5.  I  —  4  =.095         10.  4  of.yV— ^  of  yf7j=.078 

6.  14}— y6j  =13.95       11.  2^X1— f  of  ^  =  1.429 

7.  12^8J=  3.92       12.  13}— 1^X5^  =  9^1 

T 


ADDITION  OF  COMMON  FBACTIONS. 

Art.  24.     Fractions  of  the  same  denomination  an 
added  together,  hy  finding  the  sum  of  the  numerators, 

1.  1+5  _|. 7 =2+5 -1-73=14  thirds,  or  4f . 

9        64-'7_L84.9    4.2      f\        4       4_     15     _1_     98     —  1l7 

Q        5_L64_7J_8   04    fi_5  1      _L     ^'l     _L718  Qfil 

4    _3  4.  2  4.i3_i-_i  — 1  7  7  jx  4-  J_8-  4-  iio.  =1  336 


186  FRACTIONS — ADDITION. 


Art.  26.  Fractions  of  different  denominations  are 
added  together,  by  finding  a  common  denominator,  as  in 
subtraction^  and  proceeding  as  under  last  Art. 

1.  Let  it  be  required  to  add  together  f +  |+f. 

4X6X8  =  192  common  denominator. 
4)192 

^Moo  Explanation. — The  common  denomina 

6)19^  tor  was  found  as  in  subtraction,  Art.    22, 

~^'^  y  ^. ^  p/^  and  the  numerators  by  taking  the  fractional 

o^XO      ioU  pj^j-ts  of  the  common  denominator,  as  J  of 

8)192  192=48  and  2  fourths=2X 48=96,  that  is 

"^X  7=168     T^&®^^- 

2.  1+  1+  i=l^\V  6,  8J+6J+12=26.75 

3.  4+   4+  i=  tVh  ^'  2J+^Hil=  3.98 

4.  1+  f+fi=l^\  8.  1-1+9  +A=  9.61 

5.  6i+7§+8^=22/^       .    9.  64+l|+2i=10.37 

Remark. — The  process  of  adding  fractions  can  be  abridged  by 
using  the  least  common  denominator.  In  the  last  example,  35  may 
be  made  a  common  denominator:  |§,  f:^,  3^—35  —  ^5  ^^  ^•^'^' 
which  added  to  the  whole  numbers  :==:10.37. 

Art.  26.  To  find  the  least  common  denominator,  the 
process  is  as  follows:  to  add  ^+f+|+f+-^. 

2")2      4      6      7    '8  Explanation. — We  divide  all  the 

^ , denominators  by  such  a  number  as 

2)1      2      3      7      4  will  divide  into  most  of  them  with- 

out   a    remainder,     bringing    down 

3      7      2  those  that  are  not  divisible,  as  the 
2X2X3X7X2  =  168      7.     We  then  divide  the  result,  and 

■J-  of  168=    84  ^^^^  successive  result  in  the  same 

1   ((      (4    -ina  way,  until  the  numbers  can  be  re- 

6    ic      a    Z1^A(\  duced    no  lower;    after  which,   all 

I  —  i4U  ^jjg  divisors  and  the  remaining  num- 

6    "      «'    =144  bers  are  multiplied  together, 
f  <<      ^i   =147 

Til  =sQl 37 


FRACTIONS. 


187 


Bemark. — The  least  commm.  denominator  is  also  the  leaot 
COMMON  MULTIPLE,  as  it  is  tlie  least  number  which  can  be  di- 
vided by  the  several  denominators  without  a  remainder. 

4.  3J_7_l_7_L.8_4_l^!  —     9     1 

6-  T%+A+5i+8i+6  =19.91   -    ■ 

7.  2^+6|+5t+67+  f=80.9146 

8.  ^  of  f+f  of  l+f +1=2.01269 

9.  |XT93offXfofJ  +  ]=l. 14285 

10.  2iX6i+8|+f  off=25.1428 

11.  l^X2|+f  of|-ofl=  4.25 


PRACTICAL  QUESTIONS. 

1.  In  an  invoice 
items,  required  the 

of  goods, 
\  amount. 

there 

are 

the 

1  following 

27J 
16A 

doz. 
u 

@ 

u 

12 
12i 

13f 

16| 

118* 

doz. 

u 

@ 
Arts 

2* 
.  $10.65. 

Answers  will  not  be  given  to  the  following,  as 
the  pupil  can  easily  prove  the  accuracy  of  his  own 
calculations. 

2.  I  of  a  merchant's  goods  were  destroyed  by  fire, 
and  what  remained  was  worth  $1637.50,  what  was 
his  loss? 

3.  A  owns  f  of  a  steamboat,  B  J,  and  C  the  re- 
mainder, which  is  worth  $1000 ;  what  is  the  value 
of  the  boat? 

4.  J  of  a  saw  mill  belongs  to  A,  |-  to  B,  A  to  C, 
the  remainder  to  D,  and  the  profits  for  the  year 
amount   to    $1680 ;    what    is    each    man's   share  ? 


188  FRi^CTIONS — PRACTICAL    QUESTIONS. 

5.  The  par- value  of  the  pound  sterling  is  S^^^,  re- 
quired the  value  of  £1674,  at  10  %  premium. 

6.  A  can  do  a  piece  of  work  in  8  days,  E  in  7 
days,  and  C  in  6  days;  in  what  time  can  they  do  it 
if  all  work  together? 

Solution. — A  can  do  -J-,  B  ^,  and  C  -J-  of  the  work 
in  a  day.  The  sum  of  these  fractions  is  j^g.  If  ^^-^ 
can  be  done  in  a  day,  \  ||  (the  whole),  can  be  done 
in  V/=27i  or  2  days  ^  hours. 

7.  There  are  3  pumps  placed  in  a  coffer  dam ;  one 
will  empty  it  in  10,  another  in  15,  and  the  third  in 
20  hours ;  in  what  time  can  it  be  emptied  by  work- 
ing all  three  at  once?  Ans.  4j^3 hours. 

Art.  27.  To  find  fractional  parts  of  denominate 
or  compound  numbers,  as  of  pounds,  shillings  and 
and    pence;    days,  hours,  minutes  and  seconds. 

1.  Express  |  of  a  day  in  hours,  minutes,  etc. 
-3  of  a  day  is  the  same  as  ^  of  3  days. 

Days.         Hours.        Min.  Sec. 

7)3  0  0  0 

To  17  84 

Note. — As  7  is  not  contained  in  3  days,  we  reduce  them  to 
hours=72  hours,  which  divided  by  7=10  hours  and  2  left,  etc. 

2.  In  I  of  a  pound  (British  money)  how  many 
shillings  and  pence?  Ans.  £0  16s.  8d. 

3.  In  I  of  a  bushel,  how  many  pecks  quarts,  etc.? 

Ans.  3  pecks  4  quarts. 

4.  In  ^  of  a  tun  (long  weight),  how  many  hun- 
dreds, etc.?  Ans.  3  cwt.  1  qr.  9  lbs.  5^  oz. 

5.  Find  the  f  of  £167  ISs.  6d. 
First  find  \,  and  multiply  it  by  3. 


FRACTIONS.  189 


6.  f  of  41  bushels,  3  pecks,  2  quarts,  is  how  much? 

7.  /^  of  114  tuns,  8  cwt.  2  qrs.  14  lbs.  long  weight, 
is  how  much  ? 

8.  £168  18s.  8d.,  is  how  much  in  American  cur- 
rency—old standard,  4f  ?  Ans.  $750.81. 

Eeduce  18  shillings  and  8  pence  to  the  decimal  of 
a  pound.     Page  152. 

9.  Find  the  cost  of  a  draft  on  London  for  £246 
14s.  lOd.  at  9  %  premium.  Ans.  SI  195.33. 

10.  Eeduce  S1687.25  to  British  currency,  at  9  %. 

Ans,  £348  5s.  8id. 

Art.  28.  To  find  what  part  one  number  is  of  another, 
we  place  the  one  above  the  other  in  fractional  form. 

1.  3  is  what  part  of  4?  Ans.  J. 

i 

2.  f  is  what  part  of  ^?  Ans.  -  or  |. 

8 

3.  25  lbs  is  what  part  of  45  lbs.? 

4.  3  doz.  chickens    is    what   part  of  42? 

5.  1  peck  is  what  part  of  a  bushel  ? 

6.  ^§3  of  a  pound  is  what  part  of  a  penny? 

j|^  of  a  pound  is  the  same  as  ^^g  of  £2.  Reduc- 
ing these  two  pounds  to  pence,  we  have  2X20X  12= 
480.  Ans.  ||§. 

7.  |g  of  a  minute  is  what  part  of  a  day? 

Solution. — Reduction  from  a  higher  to  a  lower  denomination, 
is  performed  by  multiplication ;  therefore  reduction  from  a  lower 
to  a  higher  denomination  will  be  performed  by  division.  To  di- 
vide a  fraction  by  a  whole  number,  we  either  multiply  the  de- 
nominator or  divide  the  numerator;  in  this  case,  we  multiply  the 
denominator. 

75      1      1 


190  DUODECIMALS. 


XXXIV.    DUODECIMALS. 

Mechanics  make  most  of  their  calculations  in  feet 
and  inches  by  duodecimals. 

Art.  1.  Duodecimals  like  Decimals^  is  a  species 
of  calculation  which  enables  the  operator  to  compute 
fractional  quantities  as  whole  numbers. 

12""  fourths  make  1  third. 

\2/"  thirds    make  1  second. 

12"  seconds  m2ikQ  1  prime  or  inch. 

12'    'primes  or  in.  make  1  foot. 

1  inch      is  the  j^^  of  a  foot. 
1  second  is  the  ^l  of  an  inch,  or    j\^    of  a  foot. 
1  third     is  the  ^3^  of  a  second,     j^i^   ^^  ^  ^'^^^' 
1  fourth  is  the  j\  of  a  third,  or  Tv^^gg  of  a  foot. 
1.    Multipl}^  the   following   dimensions   together: 
10  ft.  7  in.x3  ft.  8  in.x7  ft.  9  in. 

We  commence  to  multiply  by 

the  left-hand  figure  (3),  and  write 

the  result  without  reducing  to  a 

higher  denomination.     3X10  ft. 

=  30  ft.,  and   7  in.X3  =  21   in. 

Then    multiplying  by  the  8,  we 

write  the  first  product  under  itself 

as  the  multiplier,  and  the  second 

product,  56,  one  place  further  to 

the  right.     Adding  these,  we  have 

the  product  of  two  divisors. 

Proceeding   in    the   same   way 

300  8        11  0  with    the   7   and  9  of  the   third 

dimension,  we  add   together  the 

products  and  reduce  them  to  higher  denominations,  by  which 

we  get  300  ft.  8^  11^^,  or  300/^  ft.+T¥4=300|  ft.,  nearly. 

ft.      in.     ft.       in.       ft.         in.     " 

2.  2     5X  3      4=    8 

3.  17  IX  3  4=  56 

4.  14  6X  7  8=111 

5.  21  9X14  11=324 

6.  18  8x16  ■  7=309 


10 
3 

7 
8 

30 

21 

80 

56 

30 

7 

101 
9 

56  1st  pro. 

210 

707 
270 

392 

909  504 

0 

8 

11 

4 

2 

0 

5 

3 

6 

8 

DUODECIMALS. 

191 

7. 

8. 
9. 

• 

ft. 

4 

3 

21 

in.    ft.    in.      ft.     in.       ft.        in. 
8X6     4X17     2 
9X2     6X11     0 
11X6     7X17     8 

3159     6     3     8 

10.  How  many  squares  of  flooring  in  3  rooms 
measuring  18  ft.  6  in.Xl5  ft.  8  in.,  and  what  is  the 
cost  of  laying,  at  50  cents  per  square  ? 

18X6 
15X8 


277 

6 

12 

4 

289 

10 

3 

869  6  or  869^  sq.  feet,  which  reduced 
to  squares  of  100  feet=8.695 
squares.  8.695  X  50  cts.  = 
4.347,  or  $4.35. 

11.  What  is  the  cost  of  laying  4  floors  of  the  fol- 
lowing dimensions,  at  75  cents  per  square:  18  ft.  9  in. 
Xl7ft.  3in.? 

12.  What  will  be  the  cost  of  shingling  a  roof 
which  measures  53  ft.  6  in.  long,  and  5  ft.  8  in.  from 
the  ridge  to  the  outer  edge  of  the  wall,  at  $1.50  per 
square? 

13.  The  average  breadth  of  a  board  is  1  ft.  4  in., 
and  the  length  23  ft.  9  in.,  what  number  of  feet  does 
it  contain? 

14.  How  many  solid  feet  in  a  log  measuring  as 
follows:  45  ft.  4  in. XI  ft.  6  in.Xl  ft.  3  in.? 


192  DUODECIMALS. 


15.  What  will  it  cost  to  shingle  a  roof  26  feet  long, 
rafters  14  feet,  at  ^1.25  per  square? 

16.  How  many  square  feet  of  lumber  in  a  staircase 
12  ft.  wide,  with  23  steps  8  in.  high,  and  steps  1  ft. 
2  in.  front  to  back? 

17.  How  much  will  it  cost  to  floor  a  house  of  6 
rooms,  with  ash  lumber,  ready  for  laying,  at  $5.25 
per  hundred,  and  $1.25  per  square  for  laying?  The 
rooms  measure  as  follows:  2  rooms  18  ft.  6  in.  by 
16  ft.  8  in.,  3  rooms  16  ft.  7  in.,  by  14  ft.  6  in.,  and 
one  16  ft.  6  in.  square? 

18.  How  many  squares  in  a  partition  thar  measures 
22  ft.  6  in.  long,  and  15  ft.  4  in.  high  ? 

19.  What  will  be  the  expense  of  shingling  a  roof 
120  ft.  long,  and  18  ft.  6  in.  from  the  ridge  to  the 
side  wall  of  the  house,  at  $2.50  per  square? 

20.  How  many  cubic  feet  of  timber  are  in  17  logs 
of  the  following  dimensions  :  3  logs  40  ft. X 2X2;  5 
logs  28  ft.  Xl6  in.  square,  and  the  balance  54  ft.  X  22 
in.  square? 

21.  What  is  the  cost  of  laying  3  floors  of  the  fol- 
lowing dimensions,  at  75  cents  per  square?  16  ft. 
9  in.  X 17  ft.  3  in. 

22.  What  will  be  the  cost  of  shingling  a  roof  which 
measures  62  ft.  6  in.  long,  and  8  ft.  8  in.  from  the 
ridge  to  the  outer  edge  of  the  wall,  at  $1.50  per 
square  ? 

23.  The  average  breadth  of  a  board  is  1  ft.  6  in., 
and  the  length  14  ft.  9  in.;  what  number  of  feet  does 
it  contain  ? 

Answers:  $9.09,  85  ft.,  $9.70,  31f  sq.  ft.,  $9.10, 
506  ft.,  $104.66,  $3.45,  $111,  2362J,  $22J,  $72.23, 
$90.28. 


COMPOUND   PROPORTION.  193 


XXXV.    COMPOUND  PROPORTION. 

Art.  1.  When  there  are  more  than  three  terms 
in  a  proportion,  it  is  said  to  be  compound. 

1.  If  3  men  in  5  days,  by  working  8  hours  a  day, 
dig  a  cellar  15  feet  long,  12  feet  wide,  and  7  feet  deep, 
in  how  many  days  will  2  men  dig  one  17  feet  long, 

14  feet  wide,  and  6  feet  deep,  by  working  10  hours  a 
day? 

In  this  problem,  there  are  11  terms  and  5  ratios: 
the  ratio  between  men  and  men,  that  between  hours 
and  hours ;  between  feet  and  feet  of  the  length ;  feet 
and  feet  of  the  width,  and  feet  and  feet  of  the  depth. 
In  arranging  these  terms,  we  proceed  as  in  Simple 
Proportion,  Ex.  14  page  141. 

da. 

Days  are  wanted,  write  days  as  the  men,                   2  :  3  :  :  5 

right  hand  term.  hourS,              10:8 

2    Comparing  men  with  men,  we  find  j^^gthin  ft.  15  :  17 

that  it  will  take   2  men   a  loTi^er  time  .^.    .      o.   -tn    -,  1 

to  do  the  job,  than  it  took  3  men,  so  we  Width  in  it.  1^  :  14 

write  the  greater  of  the  two  terms  (3)  depth  in  ft.    7:6 
in  the  second  place. 

3.  Comparing  hours  with  hours,  we  reason  that  it  will  take  less 
time  to  do  the  job,  by  working  10,  than  by  working  8  hours  a 
day,  so  we  write  the  smaller  number  on  the  right,  and  under  the 
second  term. 

4.  Comparing  length  with  length,  we  reason  that  it  will  take 
a  longer  time  to  dig  a  cellar  17  feet  long,  than  it  did  to  dig  one 

15  feet  long;  so  we  write  the  greater  (17)  term  under  the  second 
term. 

6.  Comparing  breadth  with  breadth,  it  will  take  a  longer  time 
to  dig  a  cellar  14  feet  wide,  than  it  did  to  dig  one  12  feet  wide; 
so  we  write  the  greater  (14)  under  the  second  term. 

6.  Comparing  depth  with  depth,  it  will  take  less  time  to  dig  a 
cellar  6  feet  deep,  than  it  did  to  dig  one  7  feet  deep;  so  we  write 
the  smaller  number  under  the  second  term. 

13 


194  COMPOUND   PROPOETION. 

The  terms  on  the  left  being  divisors  $ 


$ 

17 

U  t 
(J    2 


of  those  on   the   right,   this    statement  % 

resolves  itself  into  a  fraction,  which  can  5  ^0 
be  solved  with  great  ease  by  cancella-  0  10 
tion.  %  U 

% 

17X2 

—  6|,  or  6  days  8  hours. 

D 

The  example  may  be  reasoned  out  thus:  If  3  men  work  5 
days  8  hour.>  per  day,  that  is  equal  to  the  work  of  one  man  for 
3X5X8=120  hours;  15X12X7  feet  is  equal  to  1260  cubic 
feet,  17  X  14  X  6  feet  is  equal  to  1428  cubic  feet.  2  men  work- 
ing 10  hours  per  day  is  equal  to  1  man  working  20  hours  per 
day.  It  takes  120  hours  to  dig  1260  cubic  ieet,  hence  10^  feet 
per  hour.  In  20  hours  210  feet  can  be  dug.  1428  feet:  210 
feet=  6f  days  or  6  days  8  hours. 

2.  If  6  men  in  15  days  dig  a  trench  18  feet  long, 
7  feet  wide,  and  5  feet  deep,  in  how  many  days  will 
21  men  dig  a  trench  125  feet  long,  9  feet  wide,  and 
4  feet  deep?  JL?i^. 30.61  days. 

3.  What  is  the  interest  of  S6784  for  2  years  6 
months,  and  15  days,  at  6  %  per  annum  (365  days)? 

Statement.  $6 

days,      365     927 
dollars,  100     6784 

Arts.  $1033.77 

4.  The  interest  of  $1467  for  3  years,  4  moa.,  and 
12  days,  is  $450.72,  what  is  the  rate  per  cent?* 

5.  The  interest  on  $786.55  at  10  %  is  $176.44, 
what  is  the  time? 

6.  The  interest  of  a  certain  sum  of  monej^  for  4 
years,  2  months,  and  20  days  at  6  %  is  $100,  required 
the  principal? 

♦  The  pupil  can  prove  his  own  work  by  computing  the  interest 
by  the  method  taught  in  the  first  part  of  this  book. 


GAUGING.  195 


XXXVI.    GAUGING. 

The  process  of  finding  the  capacity  of  barrels,  etc., 
is  callen  Gauging. 

Art.  1.  To  find  the  capacity  of  a  vessel  in  the  form 
of  a  cylinder^  square  the  diameter  in  inches,  multiply 
by  the  length  in  inches,  and  the  product  by  34,  then 
point  off  four  figures  from  the  right,  and  you  have 
the  capacity  in  wine  gallons. 

1.  Find  the  capacity  in  gallons  of  a  cistern  measur- 
ing 8  feet  in  diameter  and  lO  feet  in  depth. 

Solution.— 8  ft.  =  96  inches  ;  10  ft.  ==  120  inches.  96  X 
96  X  120  X  34  =^  3760.1280,  or  3760xVVo  g^^^- 

Note. — To  find  the  capacity  in  bbls.,  divide  the  i.umber  of 
gal.  by  31 J  (the  number  of  gal.  to  a  bbl.). 

2.  Find  the  capacity,  in  gallons  and  barrels,  of  a 
cistern  measuring  10  feet  in  diameter  and  12  feet  in 
depth. 

Art.  2.  Having  the  head  and  hung  diameters,  to  find 
the  mean  diameter  add  two-thirds  of  the  difference  to 
the  head  diameter.  To  find  the  capacity  of  a  barrel  or 
cask,  ascertain  the  mean  diameter  and  proceed  to 
solve  as  under  Art.  1. 

1.  A  cask,  having  for  the  head  and  bung  diameters 
30  and  36,  and  length  40  inches,  holds  how  many 
wine  gallons? 

30  —  36  ==  6.     f  of  6  =  4.     4  +  30  =:  34  mean  diam. 
342  =  1156  X  40  X  34  =  157.2160  gallons. 

2.  Find  the  capacity  of  a  barrel  measuring  17 
inches  at  the  head,  21  inches,  bung,  and  being  2  feet 
3  inches  long. 

3.  What  is  the  capacity  of  a  barrel,  having  the 
head  diameter  36  inches,  bung  diameter  40  inches, 
and  lenorth  46  inches? 


196  GAUGING. 


Art.  3.  Having  the  top  and  bottom  diameter  of  a 
vessel  in  the  form  of  a  frustrum  of  a  cone,  to  find  the 
mean  diameter  add  half  of  the  difference  to  the 
smaller. 

1.  Find  the  capacity  in  gallons  of  a  vat,  in  the  form 
of  a  frustrum  of  a  cone,  the  diameter  at  the  top  being 

5  feet,  and  at  the  bottom  7  feet,  and  the  depth  6  feet. 

Solution.—  5  ft.  =  60  inches.     7  ft.  =  84  inches.     60  fn  m 
84=r24,  half  which  (12)  added  to  60  (the  smaller  diameter)  =* 
72  inches  mean  diameter.     72  X  72  X  72  (depth  in  inches)  X 
34,  etc. 

2.  What  is  the  capacity  in  gallons  of  a  vat,  the  top 
and  bottom  diameters  being  4  and  6  feet,  and  the 
depth  6 J  feet? 

3.  How  many  gallons  will  a  vat  hold,  measuring 

6  feet  at  the  top,  6^  feet  at  the  bottom,  and  7  feet  in 
depth? 

Art.  4.  To  find  the  number  of  gallons  of  linseed 
oil  in  a  barrel,  add  one-third  of  the  number  of  pounds 
to  the  net  weight  in  pounds  of  the  barrel,  and  divide 
the  sum  by  10  (there  are  7^  pounds  of  linseed  oil  to 
the  gallon). 

1.  How  many  gallons  of  linseed  oil  are  contained 
in  a  barrel  weighing  315  pounds  net? 

Solution.— J  of  315  =  105  +  315  =  420.  420  divided  by 
10  =  42  gallons. 

2.  Find  the  contents  in  gallons  of  a  barrel  linseed 
oil  weighing  324  pounds. 

3.  In  a  barrel  of  linseed  oil  weighing  298  pounds, 
are  how  many  gallons  ? 

Answers:  157.216  gal.,  35J  gal.,  119.369  barrels, 
3760.128  gal.,  1269.0432  gal.,  42  gal.,  43.2  gal.,  39.733 
gal.,  233.835  gal.,  7050.24  bbls.  223.82  bbls.  954.72  gaL, 
1359.74  gal,  32yV  gal. 


iisriDEx:, 


Account  Sales^ 123 

Account  Current 128 

Acre,  Hills  in  an 12 

Square  feet  to  an 8 

Acres  in  a  field 165 

Adding,  Rapid  method  of. 22 

Addition,  simple 19 

of  Compound  numbers 148 

of  Decimals 63 

of  Fractions 185 

Ad  Valorem  duties 157 

Agents,  see  Commission 62 

Aliquot  Parts  of  60 84 

of  100 55 

of  1000 55 

of  £ 352 

Aliquots,  Multiplication  by 55 

Annual  Interest 91 

Apothecaries  Weight 6 

Arithmetical  Definitions 16 

Signs 16 

Average 118 

applied  to  accounts  current..  126 

applied  to  statements 127 

applied  to  storage 129 

Discount  product  method....  133 

Interest  product  method 134 

Compound  method 136 

applied  to  account  sales 123 

Capital  to  calculate 145 

Averaging,  Methods  of 132 

Avoirdupois  Weight 6 

B 

Bank  Discount 94 

Bankruptcy 113 

Bills  of  Exchange 151 

BiLLs-Invoices 67 

Accuracy  in 67 


Credit  on 74 

Discount  oflf 73 

with  gross  weight  and  tare..    69 

receipted  by  clerk 68 

BiLL-Making,  exercises  on 76 

Bin,  to  measure 166 

Binders'  count  of  paper 13 

Bonds,  Stocks,  etc 63 

Breakage 158 

Bricklayers'  Measure 10 

British  Money 5 

Exchange 151 

Brokerage  and  Commission 62 

BusHELin  inches 166 

of  produce.  Weight  of 7 

C 

Cancellation,  Division  by 60 

Centigrade  Thermometer 14 

Change,  to  make  rapid 28 

Charcoal 11 

Circular  Measure 9 

Cistern,  to  find  contents  of 195 

Coal  Measure 10 

Coke 11 

Commercial  Weight. 6 

Commission  and  Brokerage 62 

Complex  Percentage 103 

To  find  gain  per  cent 103 

To  find  principal 104 

To  find  rate  per  cent 105,  109 

To  find  rate  of  income 105 

To  find  cost  per  cent 106 

No  find  cost  of  investment,.  106 

To  find  the  time 107,  110 

To  find  the  rate  of  gain 108 

To  find  the  amount 109 

Miscellaneous  exercises Ill 


198 


INDEX. 


Complement 27 

Compound   Numbers 148 

Proportion 194 

Interest 90 

Corporations 146 

Cubic  or  Solid  Measure 9 

Foot,  weight  of  substances..  7 
CusTOM-HousE  Entry 160 

Values  of  Currencies 159 

Customs 157 

D 

Decimal  Point,  Power  of. 63 

Decimals,  Addition  of. 53 

Subtraction  of 54 

Multiplication  of. 178 

Division  of 174 

Reduction  of 176 

Digging 11 

Discount  and  Premium 62 

Discount,  Two  kinds  of 94 

Bank  94 

True 99 

Discounting  off  Bills 73 

Discounting  Notes 94 

Discounting  lot. -Bearing  Notes  100 

Banker's  Method 100 

Broker's  Method 101 

Equitable  Method 102 

Division 36 

Long  78 

Method  of  Proof  of. 81 

Short 36 

Principles  of 80 

Short  Methods  of. 59 

of  Fractions 182 

of  Decimals 174 

by  Cancellation 60 

Draft  or  tret 158 

Dry  Measure 10 

Duodecimals 190 

Duty 64,  157 

E 
Easy  Fractions 43 


Effect  of  Coal ii 

English  Invoices lGl-164 

Equation  of  Payments 118 

Equation  of  Time 118 

Exchange 62 

Par  of 151 

Course   of 151 

British  or  Sterling 152 

Foreign 151 

French  155 

German 154 

Extensions,  exercises  on  making   77 

F 

Fahrenheit  Thermome'^er 14 

Farming 165 

Federal  Money 5 

Foreign  Exchange 151 

"        Invoice.*; ....: 161 

«'         Bill  of  Exchange  151 

"         Currencies...  159 

Fractions,  Common 171 

Value  of  a 43 

Easy 43 

Addition  of 185 

Subtraction  of 183 

Multiplication  of 179 

Division  of. 182 

Reduction  of. ...  172 

Decimals,  see  Decimals...'?,  173 

French  Money ...   5 

Exchange 155 

G 

Gain  and  Loss 65 

per  cent 6{  111 

Gas  Measure 12 

Gauging 195 

German  Money 6 

Exchange 154 

Goods,  Exercises  in  marking....  65 

Grain,  Weight  of,  per  bushel..  7 

Greatest  Common  Divisor \73 

Gross  Weight 168 


INDEX- 


199 


H 

Hay  in  a  Ton 12 

Hills  in  an  Acre 12 

Horse  Power 12 

Heating  Power  of  Fuel 11 

I 

Importers 157 

Importing 157 

Introduction 15 

Insolvency 113 

Insurance 63 

Interest 83 

Annual  91 

Compound 90 

Simple 83 

To  find  the  true 109 

To  find  the  rate  of 109 

To  find  the  time 110 

To  find  the  amount 109 

on  Investments 144 

on  cents,  How  to  reckon 84 

for  days 84 

for  months 85 

for  years 85 

Investments 63,  103 

Invoices,  British 161    to   164 

and  Bills 67 

J 

Jewish  Long  Measure 8 

Joint  Stock  Companies 146 

L 
Land,  To  lay  off  a  quantity  of...  165 
Measure ,.      9 

f.EAKAGE 158 

Least  Common  Denominator....  186 

Least  Common  Multiple 187 

Long  Division 78 

Liquid  Measure 10 

Lumber  business,  calculation  for  169 


M 

Man,  Strength  of. 12 

Marine  Measure 8 

Marking  Goods 65 

Measure,  Ale  and  Beer 10 

Bricklayers' 10 

Circular 9 

Cloth 8 

Coal 10 

Cubic  or  Solid 9 

Dry 10 

Jewish  Long 8 

Land  9 

Linear  or  Long 8 

Liquid 10 

Marine 8 

Metric,  of  capacity 10 

Metric,  Long 8 

Scripture  Long 8 

Square  or  Surface 9 

Stone 10 

Surveyors'  8 

Time 9 

Wood 9 

Men,  Average  Weight  of. 12 

Mercantile  Order 75 

Metric  System  of  Weights  and 

Measures,  Weights 6 

Long  Measure 8 

Square  Measure 9 

Cubic  Measure 9 

Measure  of  Capacity 10 

Mint  or  Troy  Weight 6 

Multiplication 28 

of  Fractions 179 

of  Decimals 173 

by  AliqHOts 55 

Method  of  Proof  of. 81 

Short  Methods  of. 55 

Principles  of. ? 32 

Table 28 


200 


INDEX. 


N 

Net  Weight,  What 158 

Notation  and  Numeration 17 

English  Method 17 

French  Mefhod 17 

Roman  Method 18 

Notes,  Discounting 94 

Form  of  Promissory 94 

Numbers,  Compound 148 

Properties  of 55 

P 

Paper,  Sizes  of 13 

Partial  Payments  92 

Partnership,  Remarks  on 143 

Calculations 143 

Int.  on   Investment.,.,. 143 

Average  capital 144 

Winding  up... 144 

PAST-Time  Tables 116 

Payments,  Equation  of. 118 

Poll  Tax 04 

Percentage 61 

Compfex 103 

Premium  and  Discount 62\ 

Printers'   Count  of  Paper 13 

Produce,  Weight  of,  per  bushel      7 

Promissory  Note,  Form  of 94 

Properties  of  Numbers 55 

Proportion,  Simple 139 

Compound ,. 194 

R 

Ratio 138 

Reaumur  Thermometer..... 14 

Reduction  of  Decimals 176 

Roman  Notation 18 

S 

Scripture  Long  Measure... 8 

Short   Division 36 

Short  Methods 55 

Signs,  Arithmetical 16 

Simple  Interest 84 

Percentage 61 

Proportion 139 


Discount 62 

Specific   Duty 157 

Tax 64 

STATUTESof  Weight  of  Bushel...       7 

Square  Measure r. , 9 

Sterling  Money 5 

Sterling  Exchange 151 

Stocks,  Bonds,  etc 63 

Stone  Measure 10 

Strength  of  Man 13 

Subtraction 25 

of  Decimals 53 

of  Fractions 183 

of  Compound  Numbers 149 

T 

Tariff 157 

Tax , 64 

Thermometers 14 

Time,  to  Reckon 83 

Measure  of 9 

Time  Table,  Gse  of. 97 

Interest 96  and  114 

Past 110 

Ton,  Weight  per 6 

Tret 158 

Troy  Weight 6 

True  Dtscouat«;rr. 99 

Interest 109 

W 
Weight  per  Bush§I^' Produce      7 

of  Cubic  Foot.......;......... 7 

Troy 6 

Commercial 6 

per  Ton 6 

Avoirdupois 6 

Metric 6 

Gross,  tare  :ind  net.     What^^isgL- 

Live  Stock  i68  ' 

Winding  up  a  Losing  Concern...  144 

Wood    Measure 9 

Wrapping  Paper 13 


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